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Properties

Label 2-637-13.9-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.617 - 0.786i$
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  + (1.16 − 2.01i)2-s + (1.15 − 1.99i)3-s + (−1.71 − 2.97i)4-s − 3.37·5-s + (−2.69 − 4.66i)6-s − 3.34·8-s + (−1.16 − 2.01i)9-s + (−3.92 + 6.80i)10-s + (−1.16 + 2.01i)11-s − 7.93·12-s + (−0.408 + 3.58i)13-s + (−3.89 + 6.74i)15-s + (−0.466 + 0.808i)16-s + (−2.72 − 4.72i)17-s − 5.43·18-s + (−3.58 − 6.20i)19-s + ⋯
L(s)  = 1  + (0.824 − 1.42i)2-s + (0.666 − 1.15i)3-s + (−0.858 − 1.48i)4-s − 1.50·5-s + (−1.09 − 1.90i)6-s − 1.18·8-s + (−0.388 − 0.673i)9-s + (−1.24 + 2.15i)10-s + (−0.351 + 0.608i)11-s − 2.29·12-s + (−0.113 + 0.993i)13-s + (−1.00 + 1.74i)15-s + (−0.116 + 0.202i)16-s + (−0.661 − 1.14i)17-s − 1.28·18-s + (−0.822 − 1.42i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.617 - 0.786i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763004 + 1.56906i\)
\(L(\frac12)\) \(\approx\) \(0.763004 + 1.56906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.408 - 3.58i)T \)
good2 \( 1 + (-1.16 + 2.01i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.15 + 1.99i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.37T + 5T^{2} \)
11 \( 1 + (1.16 - 2.01i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.72 + 4.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.58 + 6.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.22 + 5.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.22 + 7.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.05T + 31T^{2} \)
37 \( 1 + (1.52 - 2.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.468 - 0.812i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.04 - 3.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 + (-3.62 - 6.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.19 - 5.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.30 - 3.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.79 - 6.57i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.06T + 73T^{2} \)
79 \( 1 + 7.58T + 79T^{2} \)
83 \( 1 - 2.89T + 83T^{2} \)
89 \( 1 + (-6.57 + 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.77 + 3.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.37334963675115886883640711211, −9.131833836278749519188007707735, −8.323103837724361464183753096144, −7.27595283718732150154225342631, −6.71952732512075635881454491976, −4.56515465127455670268489981780, −4.44224781987785919478104501958, −2.82971545980281424348696263257, −2.34584168565053353127241613396, −0.68030151007276984111322096984, 3.38480474173710831417303755515, 3.73248049496366691386196737027, 4.64399925509776697589490132748, 5.54105787324750404658134547920, 6.68238508044801408808555433789, 7.82592208360553833677156465455, 8.238328857462553614457021615018, 8.891813141182626132236092556628, 10.39303988952864120590059028332, 10.90589391464520607817913012072

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