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Properties

Label 2-538-269.222-c1-0-8
Degree 22
Conductor 538538
Sign −0.00927−0.999i-0.00927 - 0.999i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 0.930i)2-s + (−0.0417 + 0.890i)3-s + (−0.731 − 0.681i)4-s + (−0.816 + 0.462i)5-s + (−0.813 − 0.365i)6-s + (2.09 − 1.86i)7-s + (0.902 − 0.430i)8-s + (2.19 + 0.206i)9-s + (−0.131 − 0.929i)10-s + (0.276 + 2.34i)11-s + (0.637 − 0.623i)12-s + (−2.04 + 0.288i)13-s + (0.967 + 2.63i)14-s + (−0.378 − 0.746i)15-s + (0.0702 + 0.997i)16-s + (7.35 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.259 + 0.657i)2-s + (−0.0241 + 0.514i)3-s + (−0.365 − 0.340i)4-s + (−0.365 + 0.207i)5-s + (−0.332 − 0.149i)6-s + (0.793 − 0.705i)7-s + (0.319 − 0.152i)8-s + (0.731 + 0.0688i)9-s + (−0.0416 − 0.293i)10-s + (0.0832 + 0.707i)11-s + (0.184 − 0.179i)12-s + (−0.565 + 0.0801i)13-s + (0.258 + 0.704i)14-s + (−0.0976 − 0.192i)15-s + (0.0175 + 0.249i)16-s + (1.78 + 0.210i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.00927−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00927 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((−0.00927−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00927 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.00927−0.999i-0.00927 - 0.999i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(491,⋅)\chi_{538} (491, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), −0.00927−0.999i)(2,\ 538,\ (\ :1/2),\ -0.00927 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.917334+0.925884i0.917334 + 0.925884i
L(12)L(\frac12) ≈\approx 0.917334+0.925884i0.917334 + 0.925884i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.366−0.930i)T 1 + (0.366 - 0.930i)T
269 1+(2.62+16.1i)T 1 + (2.62 + 16.1i)T
good3 1+(0.0417−0.890i)T+(−2.98−0.280i)T2 1 + (0.0417 - 0.890i)T + (-2.98 - 0.280i)T^{2}
5 1+(0.816−0.462i)T+(2.56−4.29i)T2 1 + (0.816 - 0.462i)T + (2.56 - 4.29i)T^{2}
7 1+(−2.09+1.86i)T+(0.818−6.95i)T2 1 + (-2.09 + 1.86i)T + (0.818 - 6.95i)T^{2}
11 1+(−0.276−2.34i)T+(−10.6+2.55i)T2 1 + (-0.276 - 2.34i)T + (-10.6 + 2.55i)T^{2}
13 1+(2.04−0.288i)T+(12.4−3.60i)T2 1 + (2.04 - 0.288i)T + (12.4 - 3.60i)T^{2}
17 1+(−7.35−0.866i)T+(16.5+3.94i)T2 1 + (-7.35 - 0.866i)T + (16.5 + 3.94i)T^{2}
19 1+(1.28+0.539i)T+(13.2+13.5i)T2 1 + (1.28 + 0.539i)T + (13.2 + 13.5i)T^{2}
23 1+(−2.66−0.125i)T+(22.8+2.15i)T2 1 + (-2.66 - 0.125i)T + (22.8 + 2.15i)T^{2}
29 1+(−1.03+0.619i)T+(13.7−25.5i)T2 1 + (-1.03 + 0.619i)T + (13.7 - 25.5i)T^{2}
31 1+(−0.195−0.818i)T+(−27.6+14.0i)T2 1 + (-0.195 - 0.818i)T + (-27.6 + 14.0i)T^{2}
37 1+(5.44−8.20i)T+(−14.3−34.0i)T2 1 + (5.44 - 8.20i)T + (-14.3 - 34.0i)T^{2}
41 1+(−1.77+0.697i)T+(29.9−27.9i)T2 1 + (-1.77 + 0.697i)T + (29.9 - 27.9i)T^{2}
43 1+(−1.29−2.16i)T+(−20.3+37.8i)T2 1 + (-1.29 - 2.16i)T + (-20.3 + 37.8i)T^{2}
47 1+(−1.24+0.782i)T+(20.2−42.4i)T2 1 + (-1.24 + 0.782i)T + (20.2 - 42.4i)T^{2}
53 1+(3.42+10.8i)T+(−43.4+30.3i)T2 1 + (3.42 + 10.8i)T + (-43.4 + 30.3i)T^{2}
59 1+(−0.661+0.709i)T+(−4.14−58.8i)T2 1 + (-0.661 + 0.709i)T + (-4.14 - 58.8i)T^{2}
61 1+(−0.909−4.24i)T+(−55.6+24.9i)T2 1 + (-0.909 - 4.24i)T + (-55.6 + 24.9i)T^{2}
67 1+(−3.96+3.69i)T+(4.70−66.8i)T2 1 + (-3.96 + 3.69i)T + (4.70 - 66.8i)T^{2}
71 1+(1.25+0.459i)T+(54.1+45.9i)T2 1 + (1.25 + 0.459i)T + (54.1 + 45.9i)T^{2}
73 1+(−3.42+2.90i)T+(11.9−72.0i)T2 1 + (-3.42 + 2.90i)T + (11.9 - 72.0i)T^{2}
79 1+(−2.12+8.06i)T+(−68.7−38.9i)T2 1 + (-2.12 + 8.06i)T + (-68.7 - 38.9i)T^{2}
83 1+(0.990−5.21i)T+(−77.2−30.4i)T2 1 + (0.990 - 5.21i)T + (-77.2 - 30.4i)T^{2}
89 1+(−10.9−3.73i)T+(70.5+54.3i)T2 1 + (-10.9 - 3.73i)T + (70.5 + 54.3i)T^{2}
97 1+(1.67−7.84i)T+(−88.4−39.7i)T2 1 + (1.67 - 7.84i)T + (-88.4 - 39.7i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.75069386787736362223749370857, −10.08100722470743349525328970525, −9.435343586336350996806128947568, −8.099012940554851562965158923274, −7.50220949598917778561735167131, −6.79184313475631145553889575445, −5.26948586791160770911439055853, −4.58622623929930809906604809543, −3.56287609695316158560363001934, −1.44942318490895498146264575126, 1.00865754713979383021546709577, 2.30736410498210994252815403602, 3.66219818776785355327213115827, 4.86988297559823107637271435164, 5.86983682919163999223535340060, 7.32665333002017004620038189742, 7.970972749075844979998832021201, 8.793928122158753940493026242306, 9.785154564368910236982901260960, 10.65770825685838176955161212175

Graph of the ZZ-function along the critical line