Properties

Label 2-637-91.33-c1-0-13
Degree $2$
Conductor $637$
Sign $0.714 - 0.699i$
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  + (−0.127 + 0.474i)2-s − 2.44i·3-s + (1.52 + 0.879i)4-s + (0.931 + 3.47i)5-s + (1.15 + 0.310i)6-s + (−1.30 + 1.30i)8-s − 2.98·9-s − 1.76·10-s + (1.49 − 1.49i)11-s + (2.15 − 3.72i)12-s + (0.582 + 3.55i)13-s + (8.50 − 2.27i)15-s + (1.30 + 2.26i)16-s + (−0.572 + 0.990i)17-s + (0.378 − 1.41i)18-s + (−3.22 + 3.22i)19-s + ⋯
L(s)  = 1  + (−0.0898 + 0.335i)2-s − 1.41i·3-s + (0.761 + 0.439i)4-s + (0.416 + 1.55i)5-s + (0.473 + 0.126i)6-s + (−0.461 + 0.461i)8-s − 0.993·9-s − 0.558·10-s + (0.451 − 0.451i)11-s + (0.620 − 1.07i)12-s + (0.161 + 0.986i)13-s + (2.19 − 0.588i)15-s + (0.326 + 0.565i)16-s + (−0.138 + 0.240i)17-s + (0.0892 − 0.333i)18-s + (−0.739 + 0.739i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65306 + 0.674728i\)
\(L(\frac12)\) \(\approx\) \(1.65306 + 0.674728i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.582 - 3.55i)T \)
good2 \( 1 + (0.127 - 0.474i)T + (-1.73 - i)T^{2} \)
3 \( 1 + 2.44iT - 3T^{2} \)
5 \( 1 + (-0.931 - 3.47i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.49 + 1.49i)T - 11iT^{2} \)
17 \( 1 + (0.572 - 0.990i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.22 - 3.22i)T - 19iT^{2} \)
23 \( 1 + (1.28 - 0.741i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.75 + 4.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.85 - 1.56i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.61 - 0.700i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.21 + 4.54i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.55 - 2.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.51 + 1.74i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.74 + 3.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.0 + 2.95i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 9.46iT - 61T^{2} \)
67 \( 1 + (-5.44 - 5.44i)T + 67iT^{2} \)
71 \( 1 + (-1.74 + 6.52i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.46 - 5.46i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.91 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.06 + 5.06i)T - 83iT^{2} \)
89 \( 1 + (-2.58 + 9.63i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (13.1 + 3.52i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.89264015489590409182995093597, −9.956223856612986250690193692443, −8.523039100435354656387284667653, −7.82647159505570482114734501359, −6.87943594909211119854444691522, −6.50041952359215680247193533061, −6.00840940716406176341605703306, −3.76561183822987956930833791531, −2.56745354421274278425568345886, −1.83406354969557447037558597216, 1.05616749316570067054643382977, 2.61300402605039553554156604834, 4.03824577578931623758057472154, 4.91317158652687519256663118188, 5.59428627043029823652249596339, 6.71875591490294452954033422789, 8.240383374218548154063802193144, 9.057039561535909164239880963848, 9.712797942672059441547906488778, 10.32865107774531923797440252836

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