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Properties

Label 2-637-13.3-c1-0-8
Degree $2$
Conductor $637$
Sign $-0.929 - 0.367i$
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  + (0.651 + 1.12i)2-s + (1.44 + 2.49i)3-s + (0.151 − 0.262i)4-s − 2.88·5-s + (−1.87 + 3.25i)6-s + 3·8-s + (−2.65 + 4.59i)9-s + (−1.87 − 3.25i)10-s + (2.95 + 5.11i)11-s + 0.872·12-s + (−3.31 + 1.41i)13-s + (−4.15 − 7.19i)15-s + (1.65 + 2.86i)16-s + (−0.436 + 0.755i)17-s − 6.90·18-s + (1.44 − 2.49i)19-s + ⋯
L(s)  = 1  + (0.460 + 0.797i)2-s + (0.831 + 1.44i)3-s + (0.0756 − 0.131i)4-s − 1.28·5-s + (−0.766 + 1.32i)6-s + 1.06·8-s + (−0.883 + 1.53i)9-s + (−0.593 − 1.02i)10-s + (0.890 + 1.54i)11-s + 0.251·12-s + (−0.920 + 0.391i)13-s + (−1.07 − 1.85i)15-s + (0.412 + 0.715i)16-s + (−0.105 + 0.183i)17-s − 1.62·18-s + (0.330 − 0.572i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.929 - 0.367i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.929 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396189 + 2.07871i\)
\(L(\frac12)\) \(\approx\) \(0.396189 + 2.07871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 - 1.41i)T \)
good2 \( 1 + (-0.651 - 1.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.44 - 2.49i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.88T + 5T^{2} \)
11 \( 1 + (-2.95 - 5.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.436 - 0.755i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.44 + 2.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.30 + 5.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.651 + 1.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.872T + 31T^{2} \)
37 \( 1 + (0.697 + 1.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 9.60T + 53T^{2} \)
59 \( 1 + (-3.31 + 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.88 + 4.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.76T + 73T^{2} \)
79 \( 1 - 0.605T + 79T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + (-4.32 - 7.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.88 - 6.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.79250531383465555981853067102, −9.929789031761097156703981076889, −9.322566733233334626179026948645, −8.226819865358564604672378990780, −7.43362136098085192960933063189, −6.66788936447869144061971598997, −5.08539115899838524831019224322, −4.31366968624434947174578385894, −4.02865999138294273477824972901, −2.37829516638545847841447354908, 0.950068680988019740981576199563, 2.34319591011078898335992378749, 3.41924764994475015402629261393, 3.89277214277907347779548060582, 5.67644904227778997825071169216, 7.07475470467055880207413879501, 7.55100786847014753191941300454, 8.232448319669320290585132270173, 9.042544795781123294622184580775, 10.52862917633162207505304906989

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