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Properties

Label 2-637-91.25-c1-0-12
Degree $2$
Conductor $637$
Sign $-0.465 - 0.885i$
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  + (−2.36 + 1.36i)2-s + (−0.577 + 1.00i)3-s + (2.73 − 4.74i)4-s + (−1.62 + 0.938i)5-s − 3.16i·6-s + 9.50i·8-s + (0.831 + 1.44i)9-s + (2.56 − 4.44i)10-s + (1.99 + 1.14i)11-s + (3.16 + 5.48i)12-s + (−0.574 − 3.55i)13-s − 2.17i·15-s + (−7.51 − 13.0i)16-s + (3.03 − 5.25i)17-s + (−3.93 − 2.27i)18-s + (4.46 − 2.57i)19-s + ⋯
L(s)  = 1  + (−1.67 + 0.966i)2-s + (−0.333 + 0.577i)3-s + (1.36 − 2.37i)4-s + (−0.727 + 0.419i)5-s − 1.29i·6-s + 3.36i·8-s + (0.277 + 0.480i)9-s + (0.811 − 1.40i)10-s + (0.600 + 0.346i)11-s + (0.913 + 1.58i)12-s + (−0.159 − 0.987i)13-s − 0.560i·15-s + (−1.87 − 3.25i)16-s + (0.736 − 1.27i)17-s + (−0.928 − 0.536i)18-s + (1.02 − 0.590i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.465 - 0.885i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.465 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280357 + 0.464210i\)
\(L(\frac12)\) \(\approx\) \(0.280357 + 0.464210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.574 + 3.55i)T \)
good2 \( 1 + (2.36 - 1.36i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.577 - 1.00i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.62 - 0.938i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.99 - 1.14i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.03 + 5.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.46 + 2.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.20 - 3.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.50T + 29T^{2} \)
31 \( 1 + (-3.75 - 2.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 + 1.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 + 2.17T + 43T^{2} \)
47 \( 1 + (7.71 - 4.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.05 - 7.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.18 - 3.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.00 - 6.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.00 - 0.578i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 + (-1.70 - 0.984i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.90 + 3.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.49iT - 83T^{2} \)
89 \( 1 + (9.03 - 5.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.51iT - 97T^{2} \)
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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

−10.51931967707084066525349847374, −9.825153429955709772506414627053, −9.271839818251439632564700904531, −8.052857482986045006518371117541, −7.47521335612836806934543969026, −6.87238311977662531032811130089, −5.57635028831499698156271149748, −4.83884296780615919308149447431, −2.95046186632033927310804779787, −1.01590887815568959790693404197, 0.74772750844397244072916838435, 1.67673509261718450299256919859, 3.30048944884010114291005645980, 4.21371522127191066525001828120, 6.35640754081658458156382145765, 6.98485455296138399741590367180, 8.158964527477429401445843620392, 8.399065416486652993826634449413, 9.614275132961937258211653172838, 10.06914556830247106224782390268

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