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Properties

Label 2-637-91.9-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.113 + 0.993i$
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  + (−0.5 − 0.866i)2-s + 3·3-s + (0.500 − 0.866i)4-s + (1.5 − 2.59i)5-s + (−1.5 − 2.59i)6-s − 3·8-s + 6·9-s − 3·10-s − 3·11-s + (1.50 − 2.59i)12-s + (1 + 3.46i)13-s + (4.5 − 7.79i)15-s + (0.500 + 0.866i)16-s + (−1 + 1.73i)17-s + (−3 − 5.19i)18-s + 19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 1.73·3-s + (0.250 − 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.612 − 1.06i)6-s − 1.06·8-s + 2·9-s − 0.948·10-s − 0.904·11-s + (0.433 − 0.749i)12-s + (0.277 + 0.960i)13-s + (1.16 − 2.01i)15-s + (0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s + (−0.707 − 1.22i)18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.113 + 0.993i$
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: \((2,\ 637,\ (\ :1/2),\ -0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63638 - 1.83430i\)
\(L(\frac12)\) \(\approx\) \(1.63638 - 1.83430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 3.46i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 3T + 3T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (6.5 + 11.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08993635598612210373246806177, −9.225572483665266828648821705129, −8.989526376112955629874363703719, −8.139316254233997057346155697860, −7.00336509204686665321898464106, −5.71406097845946011556786003564, −4.60003071303646567570301607707, −3.28411039031949028525051511701, −2.16984235531105985857885849370, −1.45000882721024066151316439259, 2.43417156954458916176106786594, 2.79304304217424092196289102658, 3.80813654828250170717958005537, 5.65715707973529754724028725698, 6.72426177460275896711449148052, 7.56306674795424406332456218611, 8.007832841253923214119331855791, 8.900877084746194396536859604799, 9.778685184786831876448337038634, 10.42149659983095379738553630387

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