Properties

Label 2-637-91.4-c1-0-14
Degree $2$
Conductor $637$
Sign $-0.866 - 0.498i$
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  + 1.73i·2-s + (1 + 1.73i)3-s − 0.999·4-s + (1.5 − 0.866i)5-s + (−2.99 + 1.73i)6-s + 1.73i·8-s + (−0.499 + 0.866i)9-s + (1.49 + 2.59i)10-s + (−0.999 − 1.73i)12-s + (2.5 + 2.59i)13-s + (3 + 1.73i)15-s − 5·16-s + 3·17-s + (−1.49 − 0.866i)18-s + (−3 − 1.73i)19-s + (−1.49 + 0.866i)20-s + ⋯
L(s)  = 1  + 1.22i·2-s + (0.577 + 0.999i)3-s − 0.499·4-s + (0.670 − 0.387i)5-s + (−1.22 + 0.707i)6-s + 0.612i·8-s + (−0.166 + 0.288i)9-s + (0.474 + 0.821i)10-s + (−0.288 − 0.499i)12-s + (0.693 + 0.720i)13-s + (0.774 + 0.447i)15-s − 1.25·16-s + 0.727·17-s + (−0.353 − 0.204i)18-s + (−0.688 − 0.397i)19-s + (−0.335 + 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.866 - 0.498i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.866 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548133 + 2.05340i\)
\(L(\frac12)\) \(\approx\) \(0.548133 + 2.05340i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.66iT - 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 1.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.92iT - 59T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (6 - 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.66381790877750573040675626427, −9.788635282614945774243050231080, −8.987198245281492544597431980022, −8.502794122387243407600592933971, −7.41412908195781902925890659936, −6.37001922387249817557058387479, −5.61745489411584586416106827815, −4.63166825991413077983803237833, −3.64351257075760586848634133217, −2.04822601763219732360446813545, 1.21302577905034596293648595395, 2.20316195685096880467795827332, 2.98475326617353434658611669868, 4.16025603545610142440732389776, 5.89460202440296274178236808090, 6.59480973006934592379648538054, 7.77106904571123363072722834635, 8.418908355405181381093625544004, 9.742024094585024207627542155138, 10.22591390538241287943977090604

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