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Properties

Label 2-637-91.9-c1-0-19
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: $\chi_{637} (373, \cdot )$
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

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

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