Properties

Label 2-637-13.12-c1-0-10
Degree $2$
Conductor $637$
Sign $0.524 - 0.851i$
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  + 0.332i·2-s − 1.45·3-s + 1.88·4-s − 1.44i·5-s − 0.485i·6-s + 1.29i·8-s − 0.868·9-s + 0.480·10-s + 5.95i·11-s − 2.75·12-s + (−1.88 + 3.07i)13-s + 2.11i·15-s + 3.34·16-s + 4.32·17-s − 0.288i·18-s − 1.95i·19-s + ⋯
L(s)  = 1  + 0.235i·2-s − 0.842·3-s + 0.944·4-s − 0.646i·5-s − 0.198i·6-s + 0.457i·8-s − 0.289·9-s + 0.151·10-s + 1.79i·11-s − 0.796·12-s + (−0.524 + 0.851i)13-s + 0.544i·15-s + 0.837·16-s + 1.04·17-s − 0.0680i·18-s − 0.449i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.524 - 0.851i$
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: \((2,\ 637,\ (\ :1/2),\ 0.524 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13591 + 0.634778i\)
\(L(\frac12)\) \(\approx\) \(1.13591 + 0.634778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 - 3.07i)T \)
good2 \( 1 - 0.332iT - 2T^{2} \)
3 \( 1 + 1.45T + 3T^{2} \)
5 \( 1 + 1.44iT - 5T^{2} \)
11 \( 1 - 5.95iT - 11T^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 + 1.95iT - 19T^{2} \)
23 \( 1 - 0.540T + 23T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 - 6.10iT - 31T^{2} \)
37 \( 1 + 8.02iT - 37T^{2} \)
41 \( 1 - 7.55iT - 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 - 6.26iT - 47T^{2} \)
53 \( 1 + 2.77T + 53T^{2} \)
59 \( 1 + 0.851iT - 59T^{2} \)
61 \( 1 + 6.77T + 61T^{2} \)
67 \( 1 + 0.987iT - 67T^{2} \)
71 \( 1 - 3.76iT - 71T^{2} \)
73 \( 1 + 9.13iT - 73T^{2} \)
79 \( 1 + 0.131T + 79T^{2} \)
83 \( 1 + 2.66iT - 83T^{2} \)
89 \( 1 - 9.71iT - 89T^{2} \)
97 \( 1 + 6.58iT - 97T^{2} \)
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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

−10.76759749387299831148042351354, −9.995667697773121189163208891727, −9.039591501032668971171296239828, −7.86911492181366299840391705603, −7.00687557817289414718758978119, −6.36135850891847599421596454538, −5.17028809596356484969271436820, −4.64189046275151316793589964348, −2.80176376661513841558521725495, −1.46677842796416818095654918085, 0.836801234523705902849339999878, 2.79691548136839716130964532491, 3.37218959300406712897258314511, 5.27604471252017865538041657587, 5.98351205188568006191595046481, 6.62109160248313810079433703780, 7.74698328969699066380130863980, 8.533044987116755792264629488565, 10.14548881952730059632860822464, 10.51126710954249488036774628335

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