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Properties

Label 2-650-65.34-c2-0-16
Degree $2$
Conductor $650$
Sign $-0.0277 - 0.999i$
Analytic cond. $17.7112$
Root an. cond. $4.20846$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + 4.44i·3-s − 2i·4-s + (−4.44 − 4.44i)6-s + (−0.674 − 0.674i)7-s + (2 + 2i)8-s − 10.7·9-s + (12.0 − 12.0i)11-s + 8.89·12-s + 13·13-s + 1.34·14-s − 4·16-s + 17.6·17-s + (10.7 − 10.7i)18-s + (−6 − 6i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + 1.48i·3-s − 0.5i·4-s + (−0.741 − 0.741i)6-s + (−0.0963 − 0.0963i)7-s + (0.250 + 0.250i)8-s − 1.19·9-s + (1.09 − 1.09i)11-s + 0.741·12-s + 13-s + 0.0963·14-s − 0.250·16-s + 1.04·17-s + (0.599 − 0.599i)18-s + (−0.315 − 0.315i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0277 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0277 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.0277 - 0.999i$
Analytic conductor: \(17.7112\)
Root analytic conductor: \(4.20846\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1),\ -0.0277 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.585333339\)
\(L(\frac12)\) \(\approx\) \(1.585333339\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 \)
13 \( 1 - 13T \)
good3 \( 1 - 4.44iT - 9T^{2} \)
7 \( 1 + (0.674 + 0.674i)T + 49iT^{2} \)
11 \( 1 + (-12.0 + 12.0i)T - 121iT^{2} \)
17 \( 1 - 17.6T + 289T^{2} \)
19 \( 1 + (6 + 6i)T + 361iT^{2} \)
23 \( 1 - 18.6T + 529T^{2} \)
29 \( 1 - 10.3T + 841T^{2} \)
31 \( 1 + (-18.6 - 18.6i)T + 961iT^{2} \)
37 \( 1 + (2.30 + 2.30i)T + 1.36e3iT^{2} \)
41 \( 1 + (13.3 + 13.3i)T + 1.68e3iT^{2} \)
43 \( 1 - 60.0T + 1.84e3T^{2} \)
47 \( 1 + (65.4 + 65.4i)T + 2.20e3iT^{2} \)
53 \( 1 - 50.3iT - 2.80e3T^{2} \)
59 \( 1 + (-27.4 + 27.4i)T - 3.48e3iT^{2} \)
61 \( 1 - 19.0T + 3.72e3T^{2} \)
67 \( 1 + (36.7 - 36.7i)T - 4.48e3iT^{2} \)
71 \( 1 + (-72.7 - 72.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (49.7 + 49.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 42.7T + 6.24e3T^{2} \)
83 \( 1 + (75.4 - 75.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (6.30 - 6.30i)T - 7.92e3iT^{2} \)
97 \( 1 + (2.60 - 2.60i)T - 9.40e3iT^{2} \)
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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

−10.42790034061370712153837249134, −9.652103915372478709070156306074, −8.811095313128910889408824637442, −8.410232683030904598370407748752, −6.93719757312355874655954538877, −6.00888480590822875194440996298, −5.15654216030803911754547595577, −3.99569383394204466482692748115, −3.23684897789566186612857225936, −1.00495312077349509364399048871, 1.01352165681998230817527628814, 1.74665506500339281933435954715, 3.04888388201779970463215319987, 4.35165409371747650314195927606, 5.99819511227004417705770916524, 6.71785943264832494704312472788, 7.55325993342280309463765997020, 8.286966895555823654483259935557, 9.225068733492884741021286052116, 10.05572270524052131134931393094

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