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Properties

Label 2-637-49.43-c1-0-14
Degree $2$
Conductor $637$
Sign $0.399 + 0.916i$
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  + (0.0552 − 0.0692i)2-s + (−2.67 − 1.28i)3-s + (0.443 + 1.94i)4-s + (−3.26 − 1.57i)5-s + (−0.237 + 0.114i)6-s + (−0.177 + 2.63i)7-s + (0.318 + 0.153i)8-s + (3.63 + 4.56i)9-s + (−0.289 + 0.139i)10-s + (−0.728 + 0.913i)11-s + (1.31 − 5.77i)12-s + (−0.623 + 0.781i)13-s + (0.173 + 0.158i)14-s + (6.71 + 8.42i)15-s + (−3.56 + 1.71i)16-s + (0.969 − 4.24i)17-s + ⋯
L(s)  = 1  + (0.0390 − 0.0489i)2-s + (−1.54 − 0.744i)3-s + (0.221 + 0.971i)4-s + (−1.46 − 0.703i)5-s + (−0.0968 + 0.0466i)6-s + (−0.0672 + 0.997i)7-s + (0.112 + 0.0542i)8-s + (1.21 + 1.52i)9-s + (−0.0914 + 0.0440i)10-s + (−0.219 + 0.275i)11-s + (0.380 − 1.66i)12-s + (−0.172 + 0.216i)13-s + (0.0462 + 0.0422i)14-s + (1.73 + 2.17i)15-s + (−0.890 + 0.428i)16-s + (0.235 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.399 + 0.916i$
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: \((2,\ 637,\ (\ :1/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.392172 - 0.257035i\)
\(L(\frac12)\) \(\approx\) \(0.392172 - 0.257035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.177 - 2.63i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (-0.0552 + 0.0692i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (2.67 + 1.28i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (3.26 + 1.57i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (0.728 - 0.913i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (-0.969 + 4.24i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 2.50T + 19T^{2} \)
23 \( 1 + (1.08 + 4.74i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-1.04 + 4.57i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 9.19T + 31T^{2} \)
37 \( 1 + (-1.67 + 7.34i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-9.89 - 4.76i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-11.0 + 5.31i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.59 - 3.25i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.885 + 3.87i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-12.4 + 6.01i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.364 - 1.59i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 3.69T + 67T^{2} \)
71 \( 1 + (-2.56 - 11.2i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.550 + 0.690i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 3.10T + 79T^{2} \)
83 \( 1 + (-3.92 - 4.91i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.84 + 6.07i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−11.17470228058270788252832001866, −9.444062377305940850884820845733, −8.404940043233787637914477177464, −7.54015145754384041547035000192, −7.11052630502886676360807504115, −5.84273541295355400482210933050, −4.93176389303128805180199657711, −4.02862735033689764680678234974, −2.42159101003060468417477957438, −0.43342129592887923784778720875, 0.898452693239652286593741405465, 3.52823271639801098753562243256, 4.27879225728761118014567191072, 5.30527153403299810348005042691, 6.13632133816382746793528995274, 7.07954936496463727985636051185, 7.72176701722169621679140492555, 9.414623235512717195692150246394, 10.38251921479426562919635522105, 10.77841161282970679310626541208

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