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Properties

Label 2-650-25.6-c1-0-29
Degree $2$
Conductor $650$
Sign $0.702 + 0.711i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (0.309 + 0.951i)2-s + (2.44 − 1.77i)3-s + (−0.809 + 0.587i)4-s + (0.597 − 2.15i)5-s + (2.44 + 1.77i)6-s − 0.954·7-s + (−0.809 − 0.587i)8-s + (1.90 − 5.86i)9-s + (2.23 − 0.0977i)10-s + (−0.0305 − 0.0938i)11-s + (−0.935 + 2.88i)12-s + (0.309 − 0.951i)13-s + (−0.294 − 0.907i)14-s + (−2.37 − 6.34i)15-s + (0.309 − 0.951i)16-s + (−2.18 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.41 − 1.02i)3-s + (−0.404 + 0.293i)4-s + (0.267 − 0.963i)5-s + (1.00 + 0.726i)6-s − 0.360·7-s + (−0.286 − 0.207i)8-s + (0.635 − 1.95i)9-s + (0.706 − 0.0309i)10-s + (−0.00919 − 0.0283i)11-s + (−0.270 + 0.831i)12-s + (0.0857 − 0.263i)13-s + (−0.0787 − 0.242i)14-s + (−0.612 − 1.63i)15-s + (0.0772 − 0.237i)16-s + (−0.530 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29925 - 0.961565i\)
\(L(\frac12)\) \(\approx\) \(2.29925 - 0.961565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.597 + 2.15i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-2.44 + 1.77i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.954T + 7T^{2} \)
11 \( 1 + (0.0305 + 0.0938i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (2.18 + 1.58i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0598 + 0.0434i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.52 - 7.76i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.21 + 2.33i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.92 - 2.84i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.66 - 5.11i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.18 - 3.63i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.38T + 43T^{2} \)
47 \( 1 + (-2.90 + 2.10i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.24 + 6.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.08 - 9.50i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.18 + 3.63i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.34 + 0.975i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.18 + 2.31i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.62 - 4.99i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.82 - 1.32i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.64 + 4.82i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.38 - 13.4i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (12.1 - 8.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.938502366863320176198714804878, −9.194117922392719860483489405910, −8.574226230220969830132957332886, −7.86645464065834082051089204138, −7.05731644041879321257062877393, −6.16244780058635847446191532725, −4.98623312792034481019369564183, −3.72652942146577237919816919985, −2.62769416705136021044201345963, −1.19304759386232544445419187097, 2.24849400601871685528262364222, 2.90210409981615594616777991918, 3.86178536053638679642415717866, 4.65022811375943002693239923301, 6.12204989074850106281411471927, 7.24045331018863064978112144217, 8.452584079401311676764082368713, 9.078566963617464504815869306623, 9.908959006645705491513853172438, 10.52976437560100793849460621840

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