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Properties

Label 2-637-13.12-c1-0-18
Degree $2$
Conductor $637$
Sign $0.911 - 0.410i$
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  + 2.48i·2-s − 1.67·3-s − 4.15·4-s + 0.675i·5-s − 4.15i·6-s − 5.35i·8-s − 0.193·9-s − 1.67·10-s − 4.48i·11-s + 6.96·12-s + (−3.28 + 1.48i)13-s − 1.13i·15-s + 4.96·16-s + 3.28·17-s − 0.481i·18-s − 5.21i·19-s + ⋯
L(s)  = 1  + 1.75i·2-s − 0.967·3-s − 2.07·4-s + 0.301i·5-s − 1.69i·6-s − 1.89i·8-s − 0.0646·9-s − 0.529·10-s − 1.35i·11-s + 2.00·12-s + (−0.911 + 0.410i)13-s − 0.292i·15-s + 1.24·16-s + 0.797·17-s − 0.113i·18-s − 1.19i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.911 - 0.410i$
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: \((2,\ 637,\ (\ :1/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492281 + 0.105786i\)
\(L(\frac12)\) \(\approx\) \(0.492281 + 0.105786i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.28 - 1.48i)T \)
good2 \( 1 - 2.48iT - 2T^{2} \)
3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 - 0.675iT - 5T^{2} \)
11 \( 1 + 4.48iT - 11T^{2} \)
17 \( 1 - 3.28T + 17T^{2} \)
19 \( 1 + 5.21iT - 19T^{2} \)
23 \( 1 - 4.76T + 23T^{2} \)
29 \( 1 + 9.31T + 29T^{2} \)
31 \( 1 - 1.63iT - 31T^{2} \)
37 \( 1 - 1.44iT - 37T^{2} \)
41 \( 1 + 7.92iT - 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 - 7.86iT - 47T^{2} \)
53 \( 1 + 3.15T + 53T^{2} \)
59 \( 1 + 2.54iT - 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + 7.35iT - 67T^{2} \)
71 \( 1 + 7.75iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 1.45iT - 83T^{2} \)
89 \( 1 + 7.79iT - 89T^{2} \)
97 \( 1 + 17.9iT - 97T^{2} \)
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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

−10.75808516649515613935371010209, −9.366218817336076873690344453367, −8.781686522308482948065726651589, −7.68989392903738618317124351602, −6.91721963257885735032629664699, −6.21055655650012577221849554404, −5.36306859188048973012810299913, −4.81931478374544776816971549563, −3.20789029472901558456484565075, −0.35376315368697938620362363447, 1.23055183737234014914427762653, 2.51940384551542578828819452204, 3.77321305723303121343986787842, 4.92744874377888851914935533350, 5.42399883371316029703684043398, 6.96862780874342541299253231063, 8.126034444126763267361530668908, 9.341531268683870187704950292979, 9.945715698495545082506128188284, 10.60420258965517624832756464770

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