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Properties

Label 2-637-91.74-c1-0-19
Degree $2$
Conductor $637$
Sign $0.908 + 0.418i$
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  − 0.579·2-s + (0.946 + 1.63i)3-s − 1.66·4-s + (−0.736 − 1.27i)5-s + (−0.548 − 0.950i)6-s + 2.12·8-s + (−0.289 + 0.502i)9-s + (0.427 + 0.739i)10-s + (−0.289 − 0.502i)11-s + (−1.57 − 2.72i)12-s + (−0.128 − 3.60i)13-s + (1.39 − 2.41i)15-s + 2.09·16-s − 1.19·17-s + (0.168 − 0.291i)18-s + (0.230 − 0.399i)19-s + ⋯
L(s)  = 1  − 0.409·2-s + (0.546 + 0.946i)3-s − 0.831·4-s + (−0.329 − 0.570i)5-s + (−0.223 − 0.387i)6-s + 0.751·8-s + (−0.0966 + 0.167i)9-s + (0.135 + 0.233i)10-s + (−0.0874 − 0.151i)11-s + (−0.454 − 0.787i)12-s + (−0.0357 − 0.999i)13-s + (0.359 − 0.623i)15-s + 0.523·16-s − 0.290·17-s + (0.0396 − 0.0686i)18-s + (0.0528 − 0.0915i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03007 - 0.225931i\)
\(L(\frac12)\) \(\approx\) \(1.03007 - 0.225931i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.128 + 3.60i)T \)
good2 \( 1 + 0.579T + 2T^{2} \)
3 \( 1 + (-0.946 - 1.63i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.736 + 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.289 + 0.502i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 + (-0.230 + 0.399i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.36T + 23T^{2} \)
29 \( 1 + (-3.44 + 5.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.16T + 37T^{2} \)
41 \( 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.02 + 6.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.75 - 9.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.69 + 8.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.240T + 59T^{2} \)
61 \( 1 + (3.86 - 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.724 - 1.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.25 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.84 + 3.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.03 + 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 2.49T + 89T^{2} \)
97 \( 1 + (7.82 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.18959329775372536320392787884, −9.654977590258728194680102112850, −8.676126221932964059805741573621, −8.409329335751048881566816964427, −7.30773792968958589999772821328, −5.73083036052229599421266503381, −4.68330006120752527351014588015, −4.09840852892491596434006468933, −2.93763734508302235611660294863, −0.74268881960285116168918604274, 1.32204334879384365880013536142, 2.67339105952063251936383461332, 3.99895295430819206552498111960, 5.03406990273487663456892356921, 6.54306378088053582098632234821, 7.29700966037458428552247362415, 7.977737751710605754359507400566, 8.843305502601304640024906774023, 9.542638295138275149120493460150, 10.60137771255938385272767012646

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