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Properties

Label 2-637-13.12-c1-0-33
Degree $2$
Conductor $637$
Sign $-0.698 - 0.715i$
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  − 2.12i·2-s − 0.178·3-s − 2.51·4-s + 3.60i·5-s + 0.380i·6-s + 1.10i·8-s − 2.96·9-s + 7.66·10-s − 3.99i·11-s + 0.450·12-s + (−2.51 − 2.58i)13-s − 0.644i·15-s − 2.69·16-s − 4.78·17-s + 6.30i·18-s − 3.15i·19-s + ⋯
L(s)  = 1  − 1.50i·2-s − 0.103·3-s − 1.25·4-s + 1.61i·5-s + 0.155i·6-s + 0.389i·8-s − 0.989·9-s + 2.42·10-s − 1.20i·11-s + 0.129·12-s + (−0.698 − 0.715i)13-s − 0.166i·15-s − 0.674·16-s − 1.16·17-s + 1.48i·18-s − 0.722i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.698 - 0.715i$
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: \((2,\ 637,\ (\ :1/2),\ -0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152800 + 0.362521i\)
\(L(\frac12)\) \(\approx\) \(0.152800 + 0.362521i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 + 2.58i)T \)
good2 \( 1 + 2.12iT - 2T^{2} \)
3 \( 1 + 0.178T + 3T^{2} \)
5 \( 1 - 3.60iT - 5T^{2} \)
11 \( 1 + 3.99iT - 11T^{2} \)
17 \( 1 + 4.78T + 17T^{2} \)
19 \( 1 + 3.15iT - 19T^{2} \)
23 \( 1 - 2.17T + 23T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 - 1.48iT - 31T^{2} \)
37 \( 1 + 4.96iT - 37T^{2} \)
41 \( 1 + 2.11iT - 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + 1.01iT - 47T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 - 4.90iT - 59T^{2} \)
61 \( 1 + 2.03T + 61T^{2} \)
67 \( 1 - 3.91iT - 67T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + 3.08iT - 73T^{2} \)
79 \( 1 - 1.96T + 79T^{2} \)
83 \( 1 - 7.66iT - 83T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
97 \( 1 - 1.35iT - 97T^{2} \)
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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

−10.53481958292793288619792290574, −9.433003046310778478952952855962, −8.629150548815858953289811269288, −7.33243204952063687116715150781, −6.42610885507054196445667015326, −5.34699799808150904940926162642, −3.79441106287158551614691090714, −2.93616216430445423900116582820, −2.40012539105666643596393332657, −0.19957479029904661661591411224, 2.02589112340565463278156527685, 4.30642696289158454565441191315, 4.91654376617774485879099430788, 5.67275512794531184527438940260, 6.69594698535491023543106188860, 7.59922482380063654356516180051, 8.452670952756250047431941602314, 9.066103728896313392157486233357, 9.722386610404181950368108135556, 11.30961936473389013792512321446

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