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Properties

Label 2-845-65.58-c1-0-32
Degree 22
Conductor 845845
Sign 0.550−0.834i0.550 - 0.834i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  + (1.31 + 0.759i)2-s + (−0.653 + 0.175i)3-s + (0.152 + 0.263i)4-s + (2.15 + 0.600i)5-s + (−0.991 − 0.265i)6-s + (1.29 + 2.24i)7-s − 2.57i·8-s + (−2.20 + 1.27i)9-s + (2.37 + 2.42i)10-s + (4.82 − 1.29i)11-s + (−0.145 − 0.145i)12-s + 3.93i·14-s + (−1.51 − 0.0150i)15-s + (2.25 − 3.91i)16-s + (−0.0211 + 0.0790i)17-s − 3.85·18-s + ⋯
L(s)  = 1  + (0.929 + 0.536i)2-s + (−0.377 + 0.101i)3-s + (0.0761 + 0.131i)4-s + (0.963 + 0.268i)5-s + (−0.404 − 0.108i)6-s + (0.490 + 0.849i)7-s − 0.910i·8-s + (−0.733 + 0.423i)9-s + (0.751 + 0.766i)10-s + (1.45 − 0.390i)11-s + (−0.0420 − 0.0420i)12-s + 1.05i·14-s + (−0.390 − 0.00389i)15-s + (0.564 − 0.977i)16-s + (−0.00513 + 0.0191i)17-s − 0.909·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.550−0.834i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.550−0.834i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.550−0.834i0.550 - 0.834i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(188,⋅)\chi_{845} (188, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.550−0.834i)(2,\ 845,\ (\ :1/2),\ 0.550 - 0.834i)

Particular Values

L(1)L(1) ≈\approx 2.35571+1.26845i2.35571 + 1.26845i
L(12)L(\frac12) ≈\approx 2.35571+1.26845i2.35571 + 1.26845i
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)
bad5 1+(−2.15−0.600i)T 1 + (-2.15 - 0.600i)T
13 1 1
good2 1+(−1.31−0.759i)T+(1+1.73i)T2 1 + (-1.31 - 0.759i)T + (1 + 1.73i)T^{2}
3 1+(0.653−0.175i)T+(2.59−1.5i)T2 1 + (0.653 - 0.175i)T + (2.59 - 1.5i)T^{2}
7 1+(−1.29−2.24i)T+(−3.5+6.06i)T2 1 + (-1.29 - 2.24i)T + (-3.5 + 6.06i)T^{2}
11 1+(−4.82+1.29i)T+(9.52−5.5i)T2 1 + (-4.82 + 1.29i)T + (9.52 - 5.5i)T^{2}
17 1+(0.0211−0.0790i)T+(−14.7−8.5i)T2 1 + (0.0211 - 0.0790i)T + (-14.7 - 8.5i)T^{2}
19 1+(0.726−2.71i)T+(−16.4−9.5i)T2 1 + (0.726 - 2.71i)T + (-16.4 - 9.5i)T^{2}
23 1+(−1.05−3.91i)T+(−19.9+11.5i)T2 1 + (-1.05 - 3.91i)T + (-19.9 + 11.5i)T^{2}
29 1+(4.31+2.49i)T+(14.5+25.1i)T2 1 + (4.31 + 2.49i)T + (14.5 + 25.1i)T^{2}
31 1+(−2.32+2.32i)T−31iT2 1 + (-2.32 + 2.32i)T - 31iT^{2}
37 1+(−0.285+0.494i)T+(−18.5−32.0i)T2 1 + (-0.285 + 0.494i)T + (-18.5 - 32.0i)T^{2}
41 1+(−2.69−10.0i)T+(−35.5+20.5i)T2 1 + (-2.69 - 10.0i)T + (-35.5 + 20.5i)T^{2}
43 1+(0.132+0.0354i)T+(37.2+21.5i)T2 1 + (0.132 + 0.0354i)T + (37.2 + 21.5i)T^{2}
47 1−2.30T+47T2 1 - 2.30T + 47T^{2}
53 1+(−6.70−6.70i)T+53iT2 1 + (-6.70 - 6.70i)T + 53iT^{2}
59 1+(2.59+0.694i)T+(51.0+29.5i)T2 1 + (2.59 + 0.694i)T + (51.0 + 29.5i)T^{2}
61 1+(2.74+4.74i)T+(−30.5+52.8i)T2 1 + (2.74 + 4.74i)T + (-30.5 + 52.8i)T^{2}
67 1+(13.6+7.89i)T+(33.5+58.0i)T2 1 + (13.6 + 7.89i)T + (33.5 + 58.0i)T^{2}
71 1+(7.42+1.98i)T+(61.4+35.5i)T2 1 + (7.42 + 1.98i)T + (61.4 + 35.5i)T^{2}
73 1−6.61iT−73T2 1 - 6.61iT - 73T^{2}
79 1+5.71iT−79T2 1 + 5.71iT - 79T^{2}
83 1+3.70T+83T2 1 + 3.70T + 83T^{2}
89 1+(4.63+17.2i)T+(−77.0+44.5i)T2 1 + (4.63 + 17.2i)T + (-77.0 + 44.5i)T^{2}
97 1+(−4.65+2.68i)T+(48.5−84.0i)T2 1 + (-4.65 + 2.68i)T + (48.5 - 84.0i)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

−10.32388621521171108987550045741, −9.407000582145317195279407396981, −8.796502976910130941452832392531, −7.53491704604868062128816358625, −6.24314469280300030296274890466, −5.99667370497567152175019367901, −5.27221974282099532686599752958, −4.27195882641174546176685433048, −3.00604379210378082981054656107, −1.56914670491405406544045682754, 1.23736946958450352686670350882, 2.51438004034935702098593987162, 3.79714937141590995814973549097, 4.59552340167353895213159707003, 5.47324503241391808637106277448, 6.36563868551802132486744377028, 7.23441263780066662688576067453, 8.700609171972631808768935025903, 9.087425898700827798526934948957, 10.37689865774745240075309777691

Graph of the ZZ-function along the critical line