Properties

Label 2-637-49.39-c1-0-27
Degree $2$
Conductor $637$
Sign $0.269 + 0.962i$
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  + (−1.79 − 0.270i)2-s + (−0.0106 + 0.00723i)3-s + (1.23 + 0.382i)4-s + (0.0531 + 0.709i)5-s + (0.0210 − 0.0101i)6-s + (−2.62 − 0.358i)7-s + (1.15 + 0.554i)8-s + (−1.09 + 2.79i)9-s + (0.0965 − 1.28i)10-s + (−1.14 − 2.90i)11-s + (−0.0159 + 0.00490i)12-s + (0.623 − 0.781i)13-s + (4.60 + 1.35i)14-s + (−0.00569 − 0.00714i)15-s + (−4.05 − 2.76i)16-s + (−0.265 − 0.246i)17-s + ⋯
L(s)  = 1  + (−1.26 − 0.191i)2-s + (−0.00612 + 0.00417i)3-s + (0.619 + 0.191i)4-s + (0.0237 + 0.317i)5-s + (0.00858 − 0.00413i)6-s + (−0.990 − 0.135i)7-s + (0.406 + 0.195i)8-s + (−0.365 + 0.930i)9-s + (0.0305 − 0.407i)10-s + (−0.343 − 0.876i)11-s + (−0.00459 + 0.00141i)12-s + (0.172 − 0.216i)13-s + (1.23 + 0.361i)14-s + (−0.00147 − 0.00184i)15-s + (−1.01 − 0.691i)16-s + (−0.0644 − 0.0597i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.269 + 0.962i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.269 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378610 - 0.287192i\)
\(L(\frac12)\) \(\approx\) \(0.378610 - 0.287192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 + 0.358i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (1.79 + 0.270i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (0.0106 - 0.00723i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.0531 - 0.709i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (1.14 + 2.90i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (0.265 + 0.246i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-1.13 - 1.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.43 + 1.33i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.770 + 3.37i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.15 + 3.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.41 - 1.36i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (6.69 + 3.22i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-6.11 + 2.94i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-4.36 - 0.658i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (1.43 + 0.442i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-1.07 + 14.3i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-9.39 + 2.89i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-5.13 + 8.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.51 + 11.0i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-3.16 + 0.477i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (2.61 + 4.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 + 13.0i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.759 - 1.93i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 - 4.00T + 97T^{2} \)
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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

−10.44290236354855010746066570943, −9.576052890007158589176144455920, −8.662283289222416346189397395650, −8.034657247344663440374529876668, −7.11523608028603868388850272396, −6.07470661090287971546452502371, −4.96238293206036239735582969215, −3.37631235098046028746877250090, −2.29202294177056848931876144828, −0.47962113454048670420308928091, 1.08572190983978924343543461032, 2.83074360195137078087445707419, 4.15988519132028291703282647026, 5.46246602830079042732694311684, 6.80702638804739419791753731160, 7.10267503457684297589271790758, 8.528259582849073919824112476826, 8.973807428470735338989685342241, 9.739842920054089378568756589784, 10.34271990497908760808375420136

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