Properties

Label 2-637-49.39-c1-0-43
Degree $2$
Conductor $637$
Sign $-0.886 - 0.461i$
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  + (−0.915 − 0.137i)2-s + (−1.92 + 1.31i)3-s + (−1.09 − 0.337i)4-s + (−0.266 − 3.55i)5-s + (1.94 − 0.934i)6-s + (−0.713 − 2.54i)7-s + (2.62 + 1.26i)8-s + (0.882 − 2.24i)9-s + (−0.246 + 3.29i)10-s + (−1.69 − 4.32i)11-s + (2.54 − 0.784i)12-s + (0.623 − 0.781i)13-s + (0.301 + 2.43i)14-s + (5.17 + 6.49i)15-s + (−0.335 − 0.228i)16-s + (1.38 + 1.28i)17-s + ⋯
L(s)  = 1  + (−0.647 − 0.0975i)2-s + (−1.11 + 0.756i)3-s + (−0.546 − 0.168i)4-s + (−0.119 − 1.59i)5-s + (0.792 − 0.381i)6-s + (−0.269 − 0.962i)7-s + (0.926 + 0.446i)8-s + (0.294 − 0.749i)9-s + (−0.0780 + 1.04i)10-s + (−0.511 − 1.30i)11-s + (0.734 − 0.226i)12-s + (0.172 − 0.216i)13-s + (0.0804 + 0.649i)14-s + (1.33 + 1.67i)15-s + (−0.0837 − 0.0571i)16-s + (0.336 + 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.886 - 0.461i$
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: \((2,\ 637,\ (\ :1/2),\ -0.886 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0505424 + 0.206528i\)
\(L(\frac12)\) \(\approx\) \(0.0505424 + 0.206528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.713 + 2.54i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (0.915 + 0.137i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (1.92 - 1.31i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.266 + 3.55i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (1.69 + 4.32i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (-1.38 - 1.28i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (1.05 + 1.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.07 + 0.999i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (0.284 - 1.24i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.80 + 4.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.09 - 1.88i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (-0.694 - 0.334i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (5.76 - 2.77i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.272 + 0.0410i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-3.68 - 1.13i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.141 + 1.88i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (9.37 - 2.89i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (2.14 - 3.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.10 + 4.83i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-13.0 + 1.97i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (3.99 + 6.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.18 - 7.75i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.96 - 12.6i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−10.16738709406228071760215354453, −9.296621593460836635606090817090, −8.485934832183748883044633947297, −7.83385266038684566857510188007, −6.17533605003796821848815080749, −5.24318385027074617674237057398, −4.70393405700889723964641655055, −3.76957945316984440354020747706, −1.03573976846576256846019426004, −0.21408147697739868044500363348, 1.92124803205358082379053997438, 3.32655565967555712974385275427, 4.88150698031105731198786494601, 5.90403438944319309905250335792, 6.91410838099274410917298635462, 7.25969853304289783702177544094, 8.338862240768207975206463943856, 9.529977194520867800366456701367, 10.25004368477376224373588661654, 10.94722872334765218549166632417

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