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Properties

Label 2-637-91.81-c1-0-22
Degree $2$
Conductor $637$
Sign $0.993 + 0.110i$
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  + (0.134 − 0.232i)2-s + 1.14·3-s + (0.964 + 1.66i)4-s + (−1.28 − 2.21i)5-s + (0.153 − 0.265i)6-s + 1.05·8-s − 1.69·9-s − 0.686·10-s + 3.94·11-s + (1.10 + 1.90i)12-s + (3.15 + 1.74i)13-s + (−1.46 − 2.53i)15-s + (−1.78 + 3.09i)16-s + (0.392 + 0.679i)17-s + (−0.227 + 0.393i)18-s + 7.49·19-s + ⋯
L(s)  = 1  + (0.0947 − 0.164i)2-s + 0.659·3-s + (0.482 + 0.834i)4-s + (−0.572 − 0.992i)5-s + (0.0625 − 0.108i)6-s + 0.372·8-s − 0.564·9-s − 0.217·10-s + 1.18·11-s + (0.318 + 0.550i)12-s + (0.874 + 0.484i)13-s + (−0.378 − 0.654i)15-s + (−0.446 + 0.773i)16-s + (0.0952 + 0.164i)17-s + (−0.0535 + 0.0926i)18-s + 1.71·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10256 - 0.116852i\)
\(L(\frac12)\) \(\approx\) \(2.10256 - 0.116852i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 - 1.74i)T \)
good2 \( 1 + (-0.134 + 0.232i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 1.14T + 3T^{2} \)
5 \( 1 + (1.28 + 2.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
17 \( 1 + (-0.392 - 0.679i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 7.49T + 19T^{2} \)
23 \( 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.17 + 2.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.27 - 2.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.21 + 2.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.658 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.48 + 7.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 + 1.35T + 67T^{2} \)
71 \( 1 + (6.15 - 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.384 + 0.665i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + (-3.83 + 6.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.18 - 2.05i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.79705862303205479255916628044, −9.267991484368935091323252530342, −8.768593540217079824458690079728, −8.146541540915947626765274099141, −7.20778509067091064825312967349, −6.18031554966600123735391862101, −4.72354286717446626936206168784, −3.77938695102700815279686275971, −2.99022730877226756757896403161, −1.39073571632842701878859956515, 1.41306680102806652831764531541, 3.04274769886717482463794801712, 3.60753819574254890310851901149, 5.31341062796277158378853398467, 6.13791271734633170549377549826, 7.17342794707318778613847795629, 7.68479973598498888600173975747, 9.033875640800908777216968937787, 9.589601312280158847178103287551, 10.79425901372035853135569831530

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