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Properties

Label 2-637-91.5-c1-0-28
Degree $2$
Conductor $637$
Sign $0.926 - 0.376i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.312 + 1.16i)2-s + (1.96 − 1.13i)3-s + (0.466 − 0.269i)4-s + (0.837 − 0.224i)5-s + (1.94 + 1.94i)6-s + (2.17 + 2.17i)8-s + (1.08 − 1.87i)9-s + (0.523 + 0.907i)10-s + (−0.678 + 2.53i)11-s + (0.612 − 1.06i)12-s + (−0.104 − 3.60i)13-s + (1.39 − 1.39i)15-s + (−1.31 + 2.27i)16-s + (1.52 + 2.63i)17-s + (2.53 + 0.678i)18-s + (−0.142 + 0.0381i)19-s + ⋯
L(s)  = 1  + (0.221 + 0.825i)2-s + (1.13 − 0.656i)3-s + (0.233 − 0.134i)4-s + (0.374 − 0.100i)5-s + (0.793 + 0.793i)6-s + (0.767 + 0.767i)8-s + (0.361 − 0.626i)9-s + (0.165 + 0.286i)10-s + (−0.204 + 0.763i)11-s + (0.176 − 0.306i)12-s + (−0.0289 − 0.999i)13-s + (0.359 − 0.359i)15-s + (−0.328 + 0.569i)16-s + (0.369 + 0.640i)17-s + (0.597 + 0.160i)18-s + (−0.0326 + 0.00875i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.926 - 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79529 + 0.546723i\)
\(L(\frac12)\) \(\approx\) \(2.79529 + 0.546723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (0.104 + 3.60i)T \)
good2 \( 1 + (-0.312 - 1.16i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-1.96 + 1.13i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.837 + 0.224i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.678 - 2.53i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.52 - 2.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.142 - 0.0381i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.63 + 3.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.78T + 29T^{2} \)
31 \( 1 + (-2.47 + 9.25i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.76 + 0.741i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.27 + 2.27i)T + 41iT^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (-1.90 - 7.12i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.71 + 2.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.5 - 3.34i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.97 + 4.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.43 - 0.384i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.10 - 4.10i)T - 71iT^{2} \)
73 \( 1 + (9.46 + 2.53i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.79 - 15.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + (-1.24 - 4.63i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.44 - 4.44i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.39306547733256399663916088410, −9.713581929992952908859393062571, −8.487059872474353885215092111603, −7.80740472318362876225245262304, −7.35447804606208930869385687250, −6.16062681673524526767670450149, −5.48268213707139504353192376968, −4.09586804326390688123064773435, −2.58179034780285088659483187809, −1.77189895609772623972635171393, 1.79153465047313954596976523667, 2.83800957593504933566637928896, 3.59989625684400006417737128264, 4.49276381438571288395263815813, 5.94924572914099811909607916264, 7.12550820373346718265960022879, 8.100915108009763901078612644424, 8.980239868324438458842405444398, 9.854463180219342453455685650033, 10.31564286401790135115058707166

Graph of the $Z$-function along the critical line