Properties

Label 2-637-49.39-c1-0-54
Degree $2$
Conductor $637$
Sign $-0.664 + 0.747i$
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  + (1.40 + 0.211i)2-s + (0.102 − 0.0698i)3-s + (0.0118 + 0.00366i)4-s + (−0.300 − 4.01i)5-s + (0.158 − 0.0763i)6-s + (−2.12 − 1.57i)7-s + (−2.54 − 1.22i)8-s + (−1.09 + 2.77i)9-s + (0.426 − 5.69i)10-s + (1.89 + 4.81i)11-s + (0.00147 − 0.000454i)12-s + (0.623 − 0.781i)13-s + (−2.65 − 2.65i)14-s + (−0.311 − 0.390i)15-s + (−3.32 − 2.26i)16-s + (−5.69 − 5.28i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.149i)2-s + (0.0591 − 0.0403i)3-s + (0.00594 + 0.00183i)4-s + (−0.134 − 1.79i)5-s + (0.0646 − 0.0311i)6-s + (−0.804 − 0.593i)7-s + (−0.898 − 0.432i)8-s + (−0.363 + 0.926i)9-s + (0.134 − 1.79i)10-s + (0.569 + 1.45i)11-s + (0.000425 − 0.000131i)12-s + (0.172 − 0.216i)13-s + (−0.709 − 0.709i)14-s + (−0.0803 − 0.100i)15-s + (−0.831 − 0.566i)16-s + (−1.38 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.664 + 0.747i$
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: \((2,\ 637,\ (\ :1/2),\ -0.664 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549117 - 1.22336i\)
\(L(\frac12)\) \(\approx\) \(0.549117 - 1.22336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.12 + 1.57i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (-1.40 - 0.211i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-0.102 + 0.0698i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.300 + 4.01i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-1.89 - 4.81i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (5.69 + 5.28i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (1.02 + 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.63 + 3.37i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.896 + 3.92i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.31 + 4.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.738 + 0.227i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (2.99 + 1.44i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-2.98 + 1.43i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.76 - 0.566i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-7.50 - 2.31i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.0920 - 1.22i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-8.62 + 2.66i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (0.288 - 0.500i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.12 + 4.94i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.81 - 0.875i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (6.37 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.72 - 12.1i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.54 + 3.92i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 - 3.54T + 97T^{2} \)
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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

−10.08411428546507345710802237039, −9.217956011112694442855852866199, −8.776576181535197863957435808453, −7.46141288655452250567173490509, −6.53348958085556391706003178188, −5.32148491093563050700752225830, −4.55247913365005789613949353857, −4.21251752145081113585499216990, −2.45420048361228585814244100452, −0.50945906050391076850024158751, 2.63614425251730129113685945014, 3.43095711213966511168882100471, 3.85812845783027031896310896976, 5.72573666213100374841714844882, 6.32602785456725838764725372413, 6.78549146934807241303679921171, 8.567337238205873078376872050154, 8.994446773045726714043589782004, 10.24032383463701817446815085811, 11.21141070262162840333038321297

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