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Properties

Label 2-637-91.88-c1-0-10
Degree $2$
Conductor $637$
Sign $0.417 - 0.908i$
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  + (−0.713 − 0.411i)2-s + 2.66·3-s + (−0.660 − 1.14i)4-s + (−2.73 + 1.58i)5-s + (−1.89 − 1.09i)6-s + 2.73i·8-s + 4.07·9-s + 2.60·10-s + 5.94i·11-s + (−1.75 − 3.04i)12-s + (−0.0766 + 3.60i)13-s + (−7.28 + 4.20i)15-s + (−0.195 + 0.338i)16-s + (−1.34 − 2.33i)17-s + (−2.90 − 1.67i)18-s − 1.95i·19-s + ⋯
L(s)  = 1  + (−0.504 − 0.291i)2-s + 1.53·3-s + (−0.330 − 0.572i)4-s + (−1.22 + 0.707i)5-s + (−0.774 − 0.447i)6-s + 0.967i·8-s + 1.35·9-s + 0.823·10-s + 1.79i·11-s + (−0.507 − 0.879i)12-s + (−0.0212 + 0.999i)13-s + (−1.88 + 1.08i)15-s + (−0.0488 + 0.0845i)16-s + (−0.327 − 0.567i)17-s + (−0.685 − 0.395i)18-s − 0.448i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.964884 + 0.618373i\)
\(L(\frac12)\) \(\approx\) \(0.964884 + 0.618373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0766 - 3.60i)T \)
good2 \( 1 + (0.713 + 0.411i)T + (1 + 1.73i)T^{2} \)
3 \( 1 - 2.66T + 3T^{2} \)
5 \( 1 + (2.73 - 1.58i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 5.94iT - 11T^{2} \)
17 \( 1 + (1.34 + 2.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 1.95iT - 19T^{2} \)
23 \( 1 + (1.36 - 2.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.99 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.997 - 0.575i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.63 - 3.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.23 - 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.49 + 6.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.394 + 0.228i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.199 - 0.345i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.16 - 2.40i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.15T + 61T^{2} \)
67 \( 1 + 6.27iT - 67T^{2} \)
71 \( 1 + (-3.90 - 2.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.19 + 4.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.95 - 6.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.19iT - 83T^{2} \)
89 \( 1 + (3.08 + 1.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.96 + 1.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.51562073902380448260616667760, −9.618096935654296542610214483246, −9.146708465270020253137666528297, −8.220503934953489453621846937663, −7.41483462733956271371959058673, −6.79654873146598547533251150409, −4.77755458318552392112433638263, −4.07356943693264621968645270022, −2.80917700085769624832829420419, −1.82760080674323987089260965756, 0.62997777878567892157913208409, 2.90779598767480293027265940775, 3.66008273741504957814272849605, 4.37765394464758426929552693938, 6.07955505102177327249686604775, 7.58977549235314988879680375563, 8.100037086395152295265999804967, 8.440954799444248132805433801096, 9.045009722519292637876945522224, 10.10947530519312338035560476184

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