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Properties

Label 2-637-91.45-c1-0-25
Degree $2$
Conductor $637$
Sign $0.998 - 0.0525i$
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  + (−0.193 + 0.193i)2-s + (−2.17 + 1.25i)3-s + 1.92i·4-s + (0.383 − 1.43i)5-s + (0.177 − 0.661i)6-s + (−0.758 − 0.758i)8-s + (1.64 − 2.84i)9-s + (0.202 + 0.350i)10-s + (1.18 − 4.43i)11-s + (−2.41 − 4.17i)12-s + (2.80 − 2.26i)13-s + (0.960 + 3.58i)15-s − 3.55·16-s − 2.34·17-s + (0.232 + 0.866i)18-s + (1.63 − 0.438i)19-s + ⋯
L(s)  = 1  + (−0.136 + 0.136i)2-s + (−1.25 + 0.723i)3-s + 0.962i·4-s + (0.171 − 0.639i)5-s + (0.0723 − 0.270i)6-s + (−0.268 − 0.268i)8-s + (0.546 − 0.947i)9-s + (0.0639 + 0.110i)10-s + (0.358 − 1.33i)11-s + (−0.696 − 1.20i)12-s + (0.778 − 0.627i)13-s + (0.247 + 0.925i)15-s − 0.889·16-s − 0.567·17-s + (0.0547 + 0.204i)18-s + (0.375 − 0.100i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.998 - 0.0525i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.998 - 0.0525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848281 + 0.0222946i\)
\(L(\frac12)\) \(\approx\) \(0.848281 + 0.0222946i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 + 2.26i)T \)
good2 \( 1 + (0.193 - 0.193i)T - 2iT^{2} \)
3 \( 1 + (2.17 - 1.25i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.383 + 1.43i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.18 + 4.43i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 + (-1.63 + 0.438i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.79iT - 23T^{2} \)
29 \( 1 + (2.87 - 4.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.69 + 1.52i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.03 - 6.03i)T + 37iT^{2} \)
41 \( 1 + (-0.829 + 0.222i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.70 + 0.981i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.75 + 2.07i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.54 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.09 - 1.09i)T - 59iT^{2} \)
61 \( 1 + (-8.45 - 4.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.526 - 0.141i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.77 - 0.474i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.611 - 2.28i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.88 + 3.88i)T + 83iT^{2} \)
89 \( 1 + (-9.92 + 9.92i)T - 89iT^{2} \)
97 \( 1 + (0.734 - 2.73i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.81868121983040755404612822907, −9.752691501093044274188065272431, −8.677043468230475189820586993421, −8.299359918452747200048799127083, −6.79230501421826045708455348277, −6.02048081885191743971939828050, −5.09849638726663148615174083258, −4.15888511763464257026599485067, −3.10367972958140431448380548442, −0.69015237102624325272419571741, 1.16533870400395227822722055832, 2.23996320141820390875022089436, 4.26687192402282239088903788430, 5.32293986248807895914849700387, 6.28472723389164197965998110078, 6.66201446768413577707727964371, 7.58733130850623339643352425039, 9.155449698029730335927622984824, 9.860100265588038000959107103887, 10.76274949846331060742188496613

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