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Properties

Label 2-637-13.4-c1-0-32
Degree $2$
Conductor $637$
Sign $0.0651 + 0.997i$
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  + (0.900 + 0.520i)2-s + (0.384 − 0.666i)3-s + (−0.459 − 0.795i)4-s − 1.67i·5-s + (0.692 − 0.400i)6-s − 3.03i·8-s + (1.20 + 2.08i)9-s + (0.871 − 1.51i)10-s + (−0.465 − 0.268i)11-s − 0.706·12-s + (−1.96 − 3.02i)13-s + (−1.11 − 0.644i)15-s + (0.660 − 1.14i)16-s + (−2.81 − 4.87i)17-s + 2.50i·18-s + (1.74 − 1.00i)19-s + ⋯
L(s)  = 1  + (0.636 + 0.367i)2-s + (0.222 − 0.384i)3-s + (−0.229 − 0.397i)4-s − 0.749i·5-s + (0.282 − 0.163i)6-s − 1.07i·8-s + (0.401 + 0.695i)9-s + (0.275 − 0.477i)10-s + (−0.140 − 0.0810i)11-s − 0.203·12-s + (−0.544 − 0.838i)13-s + (−0.288 − 0.166i)15-s + (0.165 − 0.285i)16-s + (−0.682 − 1.18i)17-s + 0.590i·18-s + (0.400 − 0.231i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0651 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0651 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0651 + 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.0651 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36305 - 1.27699i\)
\(L(\frac12)\) \(\approx\) \(1.36305 - 1.27699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.96 + 3.02i)T \)
good2 \( 1 + (-0.900 - 0.520i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.384 + 0.666i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.67iT - 5T^{2} \)
11 \( 1 + (0.465 + 0.268i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.81 + 4.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.74 + 1.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.33 - 5.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.43 + 4.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.78iT - 31T^{2} \)
37 \( 1 + (1.19 + 0.690i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.07 + 0.620i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.63 + 2.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.79iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + (-6.99 + 4.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.88 - 3.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.31 - 4.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 0.760i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 8.26T + 79T^{2} \)
83 \( 1 - 9.42iT - 83T^{2} \)
89 \( 1 + (1.85 + 1.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.58 - 3.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15567528184694856283926876334, −9.591222955021486627353945437790, −8.543529240809447462618422407234, −7.57758550248101298073988930160, −6.83879512704145524664727892411, −5.48305314005079172422003706830, −5.05730916481408212626687613975, −4.03555722985183111525094227258, −2.45613471600715616653870847156, −0.835639290347553603545531251760, 2.17159687836918048572391607238, 3.29593141638208951485556941611, 4.09739990837099234966040230775, 4.92586735611503281992335283776, 6.37118697574095996967744547316, 7.08678949893146958879184790947, 8.356194337872060188506280407968, 8.995819223590451733982309149533, 10.14622856091441640279586811333, 10.73255717235082295434791760594

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