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Properties

Label 2-637-91.73-c1-0-31
Degree $2$
Conductor $637$
Sign $0.754 + 0.655i$
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  + (0.423 − 1.58i)2-s + (1.71 + 0.990i)3-s + (−0.587 − 0.339i)4-s + (2.77 + 0.742i)5-s + (2.29 − 2.29i)6-s + (1.52 − 1.52i)8-s + (0.462 + 0.801i)9-s + (2.34 − 4.06i)10-s + (0.894 + 3.33i)11-s + (−0.671 − 1.16i)12-s + (−2.29 − 2.78i)13-s + (4.02 + 4.02i)15-s + (−2.44 − 4.24i)16-s + (−3.22 + 5.58i)17-s + (1.46 − 0.392i)18-s + (−2.93 − 0.786i)19-s + ⋯
L(s)  = 1  + (0.299 − 1.11i)2-s + (0.990 + 0.571i)3-s + (−0.293 − 0.169i)4-s + (1.24 + 0.332i)5-s + (0.936 − 0.936i)6-s + (0.540 − 0.540i)8-s + (0.154 + 0.267i)9-s + (0.742 − 1.28i)10-s + (0.269 + 1.00i)11-s + (−0.193 − 0.335i)12-s + (−0.635 − 0.771i)13-s + (1.03 + 1.03i)15-s + (−0.612 − 1.06i)16-s + (−0.782 + 1.35i)17-s + (0.344 − 0.0924i)18-s + (−0.673 − 0.180i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.754 + 0.655i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.754 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.77515 - 1.03731i\)
\(L(\frac12)\) \(\approx\) \(2.77515 - 1.03731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 + 2.78i)T \)
good2 \( 1 + (-0.423 + 1.58i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.71 - 0.990i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.77 - 0.742i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.894 - 3.33i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.93 + 0.786i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.97 - 1.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.40T + 29T^{2} \)
31 \( 1 + (0.868 + 3.24i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.75 - 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.03 + 3.03i)T - 41iT^{2} \)
43 \( 1 - 4.48iT - 43T^{2} \)
47 \( 1 + (-1.47 + 5.51i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.72 + 9.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.86 - 0.766i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.03 + 1.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.04 - 1.88i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.31 - 8.31i)T + 71iT^{2} \)
73 \( 1 + (4.51 - 1.20i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.543 + 0.942i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.01 - 2.01i)T - 83iT^{2} \)
89 \( 1 + (1.28 - 4.79i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.20 + 3.20i)T - 97iT^{2} \)
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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

−10.22745539050136597776377470786, −9.905776755009776217268938863956, −9.185757049260484554303747791465, −8.063434855858355237554108720938, −6.91217457120050006133980298744, −5.83613052088906147406978673544, −4.43260206498033966746168876180, −3.66245036145853775320881091066, −2.41217859905908220399430582022, −1.99536075179478418809303058488, 1.81013525377026346427559200925, 2.63462039567034056087367364010, 4.42439465761186685913121269841, 5.48757375517865391971849935846, 6.26974252443467527021671803196, 7.08569737432162631533393661485, 7.900019175512288669346104419420, 8.930036380939825384667699785043, 9.255953789836615253448707624132, 10.59086695312031296916534351800

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