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Properties

Label 2-637-13.12-c1-0-32
Degree $2$
Conductor $637$
Sign $0.996 + 0.0862i$
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  + 0.688i·2-s + 2.21·3-s + 1.52·4-s − 3.21i·5-s + 1.52i·6-s + 2.42i·8-s + 1.90·9-s + 2.21·10-s − 2.68i·11-s + 3.37·12-s + (−3.59 − 0.311i)13-s − 7.11i·15-s + 1.37·16-s + 3.59·17-s + 1.31i·18-s + 8.54i·19-s + ⋯
L(s)  = 1  + 0.487i·2-s + 1.27·3-s + 0.762·4-s − 1.43i·5-s + 0.622i·6-s + 0.858i·8-s + 0.634·9-s + 0.700·10-s − 0.810i·11-s + 0.975·12-s + (−0.996 − 0.0862i)13-s − 1.83i·15-s + 0.344·16-s + 0.871·17-s + 0.309i·18-s + 1.96i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.996 + 0.0862i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.996 + 0.0862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62575 - 0.113494i\)
\(L(\frac12)\) \(\approx\) \(2.62575 - 0.113494i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 0.311i)T \)
good2 \( 1 - 0.688iT - 2T^{2} \)
3 \( 1 - 2.21T + 3T^{2} \)
5 \( 1 + 3.21iT - 5T^{2} \)
11 \( 1 + 2.68iT - 11T^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
19 \( 1 - 8.54iT - 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + 3.93iT - 37T^{2} \)
41 \( 1 + 0.755iT - 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 - 1.88iT - 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 - 7.33iT - 59T^{2} \)
61 \( 1 + 9.05T + 61T^{2} \)
67 \( 1 - 0.428iT - 67T^{2} \)
71 \( 1 - 8.98iT - 71T^{2} \)
73 \( 1 - 5.79iT - 73T^{2} \)
79 \( 1 + 4.47T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 - 5.36iT - 89T^{2} \)
97 \( 1 + 9.62iT - 97T^{2} \)
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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

−10.29748060349502491421132984095, −9.474007303268049697593897500437, −8.534542838270513247230820580017, −8.062996421039981561498908701162, −7.40882577663716352935349979024, −5.94223799179869466443996340776, −5.24155405214017147076790293919, −3.84550983473859407535910963826, −2.73634963356593528533385365846, −1.48151181388572481382849375305, 1.99214352820529198237866958438, 2.90942201973067085614059673007, 3.23684119880560077967834410851, 4.83993584534087142284875673239, 6.58153418045794723146337232060, 7.12756381699961306019344311468, 7.73488439584258491911536779021, 9.042399366989523153030017267562, 9.892265699997627177598683001712, 10.46567800283373302614169230903

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