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Properties

Label 2-637-91.30-c1-0-24
Degree $2$
Conductor $637$
Sign $0.993 + 0.113i$
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  + (1.5 − 0.866i)2-s + (0.5 − 0.866i)4-s + (3.12 + 1.80i)5-s + 1.73i·8-s − 3·9-s + 6.24·10-s − 3.46i·11-s + (3.12 + 1.80i)13-s + (2.49 + 4.33i)16-s + (3.12 − 5.40i)17-s + (−4.5 + 2.59i)18-s + 7.21i·19-s + (3.12 − 1.80i)20-s + (−2.99 − 5.19i)22-s + (−2 − 3.46i)23-s + ⋯
L(s)  = 1  + (1.06 − 0.612i)2-s + (0.250 − 0.433i)4-s + (1.39 + 0.806i)5-s + 0.612i·8-s − 9-s + 1.97·10-s − 1.04i·11-s + (0.866 + 0.499i)13-s + (0.624 + 1.08i)16-s + (0.757 − 1.31i)17-s + (−1.06 + 0.612i)18-s + 1.65i·19-s + (0.698 − 0.403i)20-s + (−0.639 − 1.10i)22-s + (−0.417 − 0.722i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.993 + 0.113i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.993 + 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.99741 - 0.170927i\)
\(L(\frac12)\) \(\approx\) \(2.99741 - 0.170927i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 - 1.80i)T \)
good2 \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 3T^{2} \)
5 \( 1 + (-3.12 - 1.80i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 + (-3.12 + 5.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 7.21iT - 19T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.12 + 1.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.24 + 3.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.24 + 3.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 + 13.8iT - 67T^{2} \)
71 \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.36 - 5.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.21iT - 83T^{2} \)
89 \( 1 + (-6.24 + 3.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.24 + 3.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75840792021458482809777692442, −9.991638927739241122881300939416, −8.897060855331437455926357464852, −8.057960099216534452154758967362, −6.47042354969362534499273867788, −5.84518494028935359643113665102, −5.23068533018641320486376213984, −3.60263426001996451256452542916, −2.96665323100623014057856891250, −1.87121257033801735848186767441, 1.45624701312338406379266520779, 3.01153593017361447647960157986, 4.39748896659390997246022065177, 5.28934887138452257283643879287, 5.87821654605124376207972268273, 6.54672213608285406676971517527, 7.87864703362731414529123840360, 8.921038719246197347642996537815, 9.669404217820615210296448211192, 10.45557833086172384883786090150

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