Properties

Label 2-637-49.43-c1-0-7
Degree $2$
Conductor $637$
Sign $0.395 - 0.918i$
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  + (−1.44 + 1.80i)2-s + (−0.133 − 0.0644i)3-s + (−0.742 − 3.25i)4-s + (−3.89 − 1.87i)5-s + (0.309 − 0.148i)6-s + (−1.21 + 2.35i)7-s + (2.78 + 1.34i)8-s + (−1.85 − 2.32i)9-s + (9.00 − 4.33i)10-s + (−2.80 + 3.51i)11-s + (−0.110 + 0.483i)12-s + (0.623 − 0.781i)13-s + (−2.50 − 5.57i)14-s + (0.400 + 0.502i)15-s + (−0.422 + 0.203i)16-s + (0.401 − 1.75i)17-s + ⋯
L(s)  = 1  + (−1.01 + 1.27i)2-s + (−0.0772 − 0.0372i)3-s + (−0.371 − 1.62i)4-s + (−1.74 − 0.839i)5-s + (0.126 − 0.0607i)6-s + (−0.458 + 0.888i)7-s + (0.985 + 0.474i)8-s + (−0.618 − 0.776i)9-s + (2.84 − 1.37i)10-s + (−0.845 + 1.06i)11-s + (−0.0318 + 0.139i)12-s + (0.172 − 0.216i)13-s + (−0.668 − 1.49i)14-s + (0.103 + 0.129i)15-s + (−0.105 + 0.0508i)16-s + (0.0972 − 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270635 + 0.178055i\)
\(L(\frac12)\) \(\approx\) \(0.270635 + 0.178055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.21 - 2.35i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (1.44 - 1.80i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.133 + 0.0644i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (3.89 + 1.87i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.80 - 3.51i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (-0.401 + 1.75i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 0.231T + 19T^{2} \)
23 \( 1 + (1.27 + 5.59i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (1.19 - 5.22i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
37 \( 1 + (0.717 - 3.14i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-6.70 - 3.22i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.174 + 0.0839i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-4.37 + 5.48i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.41 + 6.21i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-0.0947 + 0.0456i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.53 - 11.0i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 2.01T + 67T^{2} \)
71 \( 1 + (0.819 + 3.58i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (6.06 + 7.60i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 4.15T + 79T^{2} \)
83 \( 1 + (-1.05 - 1.31i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (3.27 + 4.10i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 0.135T + 97T^{2} \)
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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

−10.45904222938540871758307100603, −9.409522805441254992936380673852, −8.701494613555628476060283969155, −8.210312897467114578724307761413, −7.39708623466904909539505780117, −6.55833769968565936432190256990, −5.45955924348083343974920988689, −4.56967714759398424683758704422, −3.07246668364845813222964612038, −0.54735678831341858799055656126, 0.54605023931656554049408970111, 2.66064391746830496978259663042, 3.41069277449282758161829868858, 4.22216630943706262264408148486, 6.03062874696808885948991075798, 7.51729304834988531266390897304, 7.88700131251925184254100615895, 8.610455050468227316268180545748, 9.886824468914930924857957888381, 10.78864898671522111861728145160

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