Properties

Label 2-637-91.83-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.988 - 0.149i$
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  + (1.90 − 1.90i)2-s − 0.759i·3-s − 5.26i·4-s + (−1.78 − 1.78i)5-s + (−1.44 − 1.44i)6-s + (−6.22 − 6.22i)8-s + 2.42·9-s − 6.81·10-s + (1.52 + 1.52i)11-s − 3.99·12-s + (1.44 + 3.30i)13-s + (−1.35 + 1.35i)15-s − 13.1·16-s + 1.40·17-s + (4.61 − 4.61i)18-s + (−1.48 − 1.48i)19-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)2-s − 0.438i·3-s − 2.63i·4-s + (−0.799 − 0.799i)5-s + (−0.590 − 0.590i)6-s + (−2.19 − 2.19i)8-s + 0.807·9-s − 2.15·10-s + (0.459 + 0.459i)11-s − 1.15·12-s + (0.401 + 0.915i)13-s + (−0.350 + 0.350i)15-s − 3.29·16-s + 0.339·17-s + (1.08 − 1.08i)18-s + (−0.339 − 0.339i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201571 + 2.68080i\)
\(L(\frac12)\) \(\approx\) \(0.201571 + 2.68080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 - 3.30i)T \)
good2 \( 1 + (-1.90 + 1.90i)T - 2iT^{2} \)
3 \( 1 + 0.759iT - 3T^{2} \)
5 \( 1 + (1.78 + 1.78i)T + 5iT^{2} \)
11 \( 1 + (-1.52 - 1.52i)T + 11iT^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 + (1.48 + 1.48i)T + 19iT^{2} \)
23 \( 1 - 1.31iT - 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + (-5.14 - 5.14i)T + 31iT^{2} \)
37 \( 1 + (-1.61 - 1.61i)T + 37iT^{2} \)
41 \( 1 + (2.69 + 2.69i)T + 41iT^{2} \)
43 \( 1 + 0.437iT - 43T^{2} \)
47 \( 1 + (-5.66 + 5.66i)T - 47iT^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + (-5.52 + 5.52i)T - 59iT^{2} \)
61 \( 1 + 7.58iT - 61T^{2} \)
67 \( 1 + (-0.401 + 0.401i)T - 67iT^{2} \)
71 \( 1 + (10.7 - 10.7i)T - 71iT^{2} \)
73 \( 1 + (8.70 - 8.70i)T - 73iT^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (3.82 + 3.82i)T + 83iT^{2} \)
89 \( 1 + (0.0366 - 0.0366i)T - 89iT^{2} \)
97 \( 1 + (-9.43 - 9.43i)T + 97iT^{2} \)
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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

−10.38431834912747942600426188956, −9.568433996728525676464716136742, −8.610721127815253811453491268899, −7.18886305365618872758507560017, −6.31216027160809041773531595789, −5.02312848064048802717080416819, −4.30993277591704528003355752530, −3.64311192186177146276550531456, −2.05854297711687862085903444570, −1.06570831580011199962035083577, 3.04024584644700181962835994688, 3.79819317592921942860514877725, 4.49044925870429412735855612332, 5.70769766263709836302902858291, 6.42999298082779194756897972900, 7.43649534991606803294758684333, 7.87164538293000030210188456173, 8.950205142156911355947793364732, 10.31987410745821998392030280516, 11.26588279655789782084252375791

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