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Properties

Label 2-637-91.20-c1-0-36
Degree $2$
Conductor $637$
Sign $0.971 + 0.237i$
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  + (2.35 − 0.629i)2-s + (1.70 + 0.985i)3-s + (3.39 − 1.96i)4-s + (−0.166 + 0.166i)5-s + (4.63 + 1.24i)6-s + (3.30 − 3.30i)8-s + (0.442 + 0.766i)9-s + (−0.286 + 0.495i)10-s + (−0.740 − 2.76i)11-s + 7.72·12-s + (−2.03 + 2.97i)13-s + (−0.447 + 0.119i)15-s + (1.76 − 3.05i)16-s + (2.01 + 3.49i)17-s + (1.52 + 1.52i)18-s + (−5.30 − 1.42i)19-s + ⋯
L(s)  = 1  + (1.66 − 0.445i)2-s + (0.985 + 0.568i)3-s + (1.69 − 0.980i)4-s + (−0.0743 + 0.0743i)5-s + (1.89 + 0.506i)6-s + (1.16 − 1.16i)8-s + (0.147 + 0.255i)9-s + (−0.0904 + 0.156i)10-s + (−0.223 − 0.833i)11-s + 2.23·12-s + (−0.563 + 0.826i)13-s + (−0.115 + 0.0309i)15-s + (0.441 − 0.764i)16-s + (0.488 + 0.846i)17-s + (0.358 + 0.358i)18-s + (−1.21 − 0.326i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.63014 - 0.556970i\)
\(L(\frac12)\) \(\approx\) \(4.63014 - 0.556970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 - 2.97i)T \)
good2 \( 1 + (-2.35 + 0.629i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-1.70 - 0.985i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.166 - 0.166i)T - 5iT^{2} \)
11 \( 1 + (0.740 + 2.76i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.01 - 3.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.30 + 1.42i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.23 + 3.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.62 + 1.62i)T - 31iT^{2} \)
37 \( 1 + (-2.68 - 10.0i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.0713 + 0.266i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.91 + 2.25i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.30 + 2.30i)T + 47iT^{2} \)
53 \( 1 - 7.93T + 53T^{2} \)
59 \( 1 + (0.188 - 0.702i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.97 - 1.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.57 - 0.688i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.362 - 1.35i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.58 + 9.58i)T + 73iT^{2} \)
79 \( 1 - 3.82T + 79T^{2} \)
83 \( 1 + (-4.19 + 4.19i)T - 83iT^{2} \)
89 \( 1 + (3.33 - 0.893i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.61 - 0.701i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.65055495119534445138609553195, −9.951017095969241662903739493039, −8.794856812660075185277386280436, −8.008635025045900707478280483333, −6.54844630036696069653117923140, −5.87209310901013686061822886240, −4.53749795505112046616710291814, −3.98658024218054189606658466464, −3.01998563265648458727754257309, −2.14691010878075244716769857469, 2.19063998138080432402563987681, 2.93616456533778366041182463349, 4.09340488547657810086289641329, 5.03274606757587068873427865758, 5.97067163573722203432846427524, 7.11712931861295375440880283375, 7.63072327876390803396335373631, 8.472123725949248843873296766904, 9.760076515043432321204509634510, 10.79016476529051053996086415785

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