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Properties

Label 2-637-7.4-c1-0-13
Degree $2$
Conductor $637$
Sign $0.605 + 0.795i$
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  + (−0.933 − 1.61i)2-s + (−1.67 + 2.90i)3-s + (−0.741 + 1.28i)4-s + (0.433 + 0.750i)5-s + 6.24·6-s − 0.965·8-s + (−4.10 − 7.11i)9-s + (0.808 − 1.39i)10-s + (1.93 − 3.34i)11-s + (−2.48 − 4.30i)12-s − 13-s − 2.90·15-s + (2.38 + 4.12i)16-s + (−1.67 + 2.90i)17-s + (−7.66 + 13.2i)18-s + (−2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + (−0.659 − 1.14i)2-s + (−0.966 + 1.67i)3-s + (−0.370 + 0.642i)4-s + (0.193 + 0.335i)5-s + 2.55·6-s − 0.341·8-s + (−1.36 − 2.37i)9-s + (0.255 − 0.442i)10-s + (0.582 − 1.00i)11-s + (−0.716 − 1.24i)12-s − 0.277·13-s − 0.748·15-s + (0.595 + 1.03i)16-s + (−0.406 + 0.703i)17-s + (−1.80 + 3.12i)18-s + (−0.617 − 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558379 - 0.276803i\)
\(L(\frac12)\) \(\approx\) \(0.558379 - 0.276803i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + (0.933 + 1.61i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.67 - 2.90i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.433 - 0.750i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.93 + 3.34i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.67 - 2.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.62 - 4.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + (-3.78 + 6.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.41 - 4.18i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.06T + 41T^{2} \)
43 \( 1 - 4.03T + 43T^{2} \)
47 \( 1 + (1.82 + 3.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.107 + 0.186i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.39 - 2.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.51 - 7.82i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.83 + 6.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.90T + 71T^{2} \)
73 \( 1 + (-7.77 + 13.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.71 + 8.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.09T + 83T^{2} \)
89 \( 1 + (0.209 + 0.362i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.11T + 97T^{2} \)
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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

−10.57854538336156087723174121776, −9.808269438410263241105193237312, −9.169945780335559196990219111927, −8.496625317591681759246469725980, −6.46527398635974966112049301849, −5.90952704591836929010279592334, −4.64691595290980830789121951508, −3.70627955530079397523053057805, −2.73043287485785794978781421092, −0.61706031465123877403041645720, 1.00785018681790447091423011225, 2.39634611083228809038662027370, 4.81116395692576187879968197717, 5.68603173571458807222172193263, 6.69503685015007181611022342004, 6.91027664995628582688884658132, 7.84735429004239104682497785957, 8.571889387278216155333144564731, 9.563386097196786302463286413794, 10.77238161355547631156170490771

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