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Properties

Label 2-637-91.4-c1-0-13
Degree $2$
Conductor $637$
Sign $-0.964 - 0.263i$
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  + 1.38i·2-s + (1.41 + 2.44i)3-s + 0.0791·4-s + (0.449 − 0.259i)5-s + (−3.39 + 1.95i)6-s + 2.88i·8-s + (−2.49 + 4.31i)9-s + (0.359 + 0.622i)10-s + (−1.40 + 0.812i)11-s + (0.111 + 0.193i)12-s + (1.42 − 3.31i)13-s + (1.26 + 0.733i)15-s − 3.83·16-s − 1.94·17-s + (−5.98 − 3.45i)18-s + (−2.15 − 1.24i)19-s + ⋯
L(s)  = 1  + 0.980i·2-s + (0.815 + 1.41i)3-s + 0.0395·4-s + (0.200 − 0.116i)5-s + (−1.38 + 0.799i)6-s + 1.01i·8-s + (−0.830 + 1.43i)9-s + (0.113 + 0.196i)10-s + (−0.424 + 0.244i)11-s + (0.0322 + 0.0559i)12-s + (0.395 − 0.918i)13-s + (0.327 + 0.189i)15-s − 0.958·16-s − 0.472·17-s + (−1.41 − 0.814i)18-s + (−0.494 − 0.285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.964 - 0.263i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.964 - 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285845 + 2.12710i\)
\(L(\frac12)\) \(\approx\) \(0.285845 + 2.12710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 3.31i)T \)
good2 \( 1 - 1.38iT - 2T^{2} \)
3 \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.449 + 0.259i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.40 - 0.812i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.94T + 17T^{2} \)
19 \( 1 + (2.15 + 1.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.14T + 23T^{2} \)
29 \( 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.01 + 2.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 + (-3.64 - 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.91 + 2.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.44 + 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.20iT - 59T^{2} \)
61 \( 1 + (-6.73 + 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.25 + 4.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.50 - 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.2 + 5.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.491 + 0.850i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.91iT - 83T^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 + (-3.82 + 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.81194916166591384677444889891, −9.962371294492806244931109453604, −9.069362792525211195779345996906, −8.409783446461402680999457262637, −7.62725792698524095111406901021, −6.51105157422266154189220429184, −5.34537495632775056383939428992, −4.76907556040422341585964297121, −3.44082843879742382614576330610, −2.42198744320962816878833523516, 1.13272787588877137482176456384, 2.18555782872539672204316516091, 2.89910144095185311192409752383, 4.10247615975847185826337187328, 5.89420040353935538341907445209, 6.94316889882362163881654057156, 7.30228162294964639463400837271, 8.702961328369168420874837544886, 9.077438245078737323311885214584, 10.44787636066128208205353356021

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