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Properties

Label 2-693-7.4-c1-0-7
Degree 22
Conductor 693693
Sign −0.934+0.354i-0.934 + 0.354i
Analytic cond. 5.533635.53363
Root an. cond. 2.352362.35236
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  + (1.39 + 2.40i)2-s + (−2.86 + 4.96i)4-s + (0.412 + 0.715i)5-s + (2.63 + 0.257i)7-s − 10.3·8-s + (−1.14 + 1.98i)10-s + (−0.5 + 0.866i)11-s − 0.296·13-s + (3.04 + 6.70i)14-s + (−8.72 − 15.1i)16-s + (−3.34 + 5.79i)17-s + (1.41 + 2.45i)19-s − 4.73·20-s − 2.78·22-s + (−1.98 − 3.43i)23-s + ⋯
L(s)  = 1  + (0.983 + 1.70i)2-s + (−1.43 + 2.48i)4-s + (0.184 + 0.319i)5-s + (0.995 + 0.0971i)7-s − 3.67·8-s + (−0.363 + 0.629i)10-s + (−0.150 + 0.261i)11-s − 0.0823·13-s + (0.813 + 1.79i)14-s + (−2.18 − 3.77i)16-s + (−0.811 + 1.40i)17-s + (0.325 + 0.562i)19-s − 1.05·20-s − 0.593·22-s + (−0.413 − 0.716i)23-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.934+0.354i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+1/2) L(s)=((−0.934+0.354i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.934+0.354i-0.934 + 0.354i
Analytic conductor: 5.533635.53363
Root analytic conductor: 2.352362.35236
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ693(298,⋅)\chi_{693} (298, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :1/2), −0.934+0.354i)(2,\ 693,\ (\ :1/2),\ -0.934 + 0.354i)

Particular Values

L(1)L(1) ≈\approx 0.408783−2.22805i0.408783 - 2.22805i
L(12)L(\frac12) ≈\approx 0.408783−2.22805i0.408783 - 2.22805i
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)
bad3 1 1
7 1+(−2.63−0.257i)T 1 + (-2.63 - 0.257i)T
11 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(−1.39−2.40i)T+(−1+1.73i)T2 1 + (-1.39 - 2.40i)T + (-1 + 1.73i)T^{2}
5 1+(−0.412−0.715i)T+(−2.5+4.33i)T2 1 + (-0.412 - 0.715i)T + (-2.5 + 4.33i)T^{2}
13 1+0.296T+13T2 1 + 0.296T + 13T^{2}
17 1+(3.34−5.79i)T+(−8.5−14.7i)T2 1 + (3.34 - 5.79i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.41−2.45i)T+(−9.5+16.4i)T2 1 + (-1.41 - 2.45i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.98+3.43i)T+(−11.5+19.9i)T2 1 + (1.98 + 3.43i)T + (-11.5 + 19.9i)T^{2}
29 1−0.484T+29T2 1 - 0.484T + 29T^{2}
31 1+(−3.66+6.34i)T+(−15.5−26.8i)T2 1 + (-3.66 + 6.34i)T + (-15.5 - 26.8i)T^{2}
37 1+(−2.86−4.96i)T+(−18.5+32.0i)T2 1 + (-2.86 - 4.96i)T + (-18.5 + 32.0i)T^{2}
41 1+0.645T+41T2 1 + 0.645T + 41T^{2}
43 1−6.43T+43T2 1 - 6.43T + 43T^{2}
47 1+(−3.86−6.70i)T+(−23.5+40.7i)T2 1 + (-3.86 - 6.70i)T + (-23.5 + 40.7i)T^{2}
53 1+(−3.55+6.16i)T+(−26.5−45.8i)T2 1 + (-3.55 + 6.16i)T + (-26.5 - 45.8i)T^{2}
59 1+(0.578−1.00i)T+(−29.5−51.0i)T2 1 + (0.578 - 1.00i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.63+4.56i)T+(−30.5+52.8i)T2 1 + (2.63 + 4.56i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.50−2.61i)T+(−33.5−58.0i)T2 1 + (1.50 - 2.61i)T + (-33.5 - 58.0i)T^{2}
71 1+3.58T+71T2 1 + 3.58T + 71T^{2}
73 1+(8.01−13.8i)T+(−36.5−63.2i)T2 1 + (8.01 - 13.8i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.16−3.74i)T+(−39.5+68.4i)T2 1 + (-2.16 - 3.74i)T + (-39.5 + 68.4i)T^{2}
83 1+2.37T+83T2 1 + 2.37T + 83T^{2}
89 1+(−6.08−10.5i)T+(−44.5+77.0i)T2 1 + (-6.08 - 10.5i)T + (-44.5 + 77.0i)T^{2}
97 1+10.7T+97T2 1 + 10.7T + 97T^{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

−11.11390394757734275769967758730, −9.896160185657778554299653381094, −8.588822972782976926706951653211, −8.185367181857123867884902036279, −7.32860099601726574321884091443, −6.33803480799238588702481836031, −5.77371767210510929930891657209, −4.60639343145168684996990703334, −4.08827073955024393522000824196, −2.54335474407397683004359898698, 0.920240972329514785461180459295, 2.14157014486405603342395612558, 3.15325825105246043716377920074, 4.43353051433278972174826423443, 5.00257846002199437642452315961, 5.80751172000778127262183021381, 7.28050172727516334823927558483, 8.837964587790223466385416977495, 9.268510863271673956782228430145, 10.37640468014159833524882417491

Graph of the ZZ-function along the critical line