Properties

Label 2-637-7.2-c1-0-27
Degree $2$
Conductor $637$
Sign $0.386 + 0.922i$
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  + (1.00 − 1.73i)2-s + (0.879 + 1.52i)3-s + (−1.01 − 1.76i)4-s + (0.452 − 0.784i)5-s + 3.53·6-s − 0.0686·8-s + (−0.0471 + 0.0816i)9-s + (−0.909 − 1.57i)10-s + (−0.358 − 0.620i)11-s + (1.78 − 3.09i)12-s − 13-s + 1.59·15-s + (1.96 − 3.40i)16-s + (1.17 + 2.03i)17-s + (0.0946 + 0.163i)18-s + (3.31 − 5.74i)19-s + ⋯
L(s)  = 1  + (0.710 − 1.22i)2-s + (0.507 + 0.879i)3-s + (−0.508 − 0.880i)4-s + (0.202 − 0.350i)5-s + 1.44·6-s − 0.0242·8-s + (−0.0157 + 0.0272i)9-s + (−0.287 − 0.498i)10-s + (−0.107 − 0.187i)11-s + (0.516 − 0.894i)12-s − 0.277·13-s + 0.411·15-s + (0.491 − 0.850i)16-s + (0.285 + 0.494i)17-s + (0.0223 + 0.0386i)18-s + (0.761 − 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27332 - 1.51217i\)
\(L(\frac12)\) \(\approx\) \(2.27332 - 1.51217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.00 + 1.73i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.879 - 1.52i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.452 + 0.784i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.358 + 0.620i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.17 - 2.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.31 + 5.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.87 - 3.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 + (-0.785 - 1.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.60 - 4.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + (4.15 - 7.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.04 + 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.358 - 0.620i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.82 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.69 + 8.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + (1.73 + 3.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.50 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.54T + 83T^{2} \)
89 \( 1 + (-6.02 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.43T + 97T^{2} \)
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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

−10.39396040338633296739688652781, −9.788527589837654874152314674203, −9.094409234727849613504451541205, −8.040110281354282338678460846185, −6.75569998060139762929586639132, −5.20198321560263242248836096524, −4.69173109060668845344392102506, −3.52550114203407585347614484885, −2.94369420165472862834769065324, −1.42547674119307034935799904636, 1.74906559064441453231161549354, 3.11355390788155762726805278217, 4.49957679171323766236930494145, 5.45780161900044374934980848106, 6.46488466701552796605253706885, 7.07021952985835325621547539190, 7.88504699018691886939264349084, 8.413466354348507262749379061020, 9.872271877764100088094803760619, 10.58864787564419623169399464629

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