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Properties

Label 2-637-7.4-c1-0-15
Degree $2$
Conductor $637$
Sign $-0.605 - 0.795i$
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  + (0.605 + 1.04i)2-s + (−0.872 + 1.51i)3-s + (0.267 − 0.462i)4-s + (1.10 + 1.91i)5-s − 2.11·6-s + 3.06·8-s + (−0.0222 − 0.0384i)9-s + (−1.33 + 2.31i)10-s + (0.394 − 0.683i)11-s + (0.465 + 0.807i)12-s + 13-s − 3.85·15-s + (1.32 + 2.29i)16-s + (−0.872 + 1.51i)17-s + (0.0268 − 0.0465i)18-s + (2.16 + 3.74i)19-s + ⋯
L(s)  = 1  + (0.428 + 0.741i)2-s + (−0.503 + 0.872i)3-s + (0.133 − 0.231i)4-s + (0.494 + 0.856i)5-s − 0.862·6-s + 1.08·8-s + (−0.00740 − 0.0128i)9-s + (−0.423 + 0.733i)10-s + (0.118 − 0.206i)11-s + (0.134 + 0.232i)12-s + 0.277·13-s − 0.995·15-s + (0.330 + 0.573i)16-s + (−0.211 + 0.366i)17-s + (0.00633 − 0.0109i)18-s + (0.495 + 0.858i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863513 + 1.74191i\)
\(L(\frac12)\) \(\approx\) \(0.863513 + 1.74191i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (-0.605 - 1.04i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.872 - 1.51i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.394 + 0.683i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.872 - 1.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 - 3.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.556 - 0.963i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-2.85 + 4.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 + 1.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - 8.06T + 43T^{2} \)
47 \( 1 + (-4.37 - 7.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.97 - 6.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.47 + 9.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.53 + 11.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.27 - 5.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.85T + 71T^{2} \)
73 \( 1 + (-4.00 + 6.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.45 - 5.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.14T + 83T^{2} \)
89 \( 1 + (-1.69 - 2.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.0981T + 97T^{2} \)
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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

−10.90391527646709271476664276981, −10.10001724661199182003023669768, −9.483387305758148563871513835442, −7.997065527556914409427668269666, −7.12641492194455636591173565909, −6.09471675877314802475407713192, −5.67122503739849240914972191275, −4.61416621282429080495741762374, −3.58972026829573545022849499640, −1.91307008976626801927001923979, 1.08210350213104928818412929754, 2.08332049483295383558027868615, 3.47822152447108134984713502529, 4.70998185669572303304709404730, 5.56761446432212740741097057825, 6.81335297347388348175700438509, 7.38307217759669385539852075185, 8.604753431381280732732916535243, 9.450651876850462695559479060784, 10.52992147417757560736646658669

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