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Properties

Label 2-304-19.18-c4-0-17
Degree $2$
Conductor $304$
Sign $-0.541 - 0.840i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.3i·3-s + 41.6·5-s + 62.4·7-s − 154.·9-s − 122.·11-s + 68.9i·13-s + 638. i·15-s + 297.·17-s + (195. + 303. i)19-s + 958. i·21-s − 268.·23-s + 1.10e3·25-s − 1.12e3i·27-s − 561. i·29-s + 252. i·31-s + ⋯
L(s)  = 1  + 1.70i·3-s + 1.66·5-s + 1.27·7-s − 1.90·9-s − 1.01·11-s + 0.408i·13-s + 2.83i·15-s + 1.02·17-s + (0.541 + 0.840i)19-s + 2.17i·21-s − 0.507·23-s + 1.77·25-s − 1.54i·27-s − 0.667i·29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.541 - 0.840i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.897404856\)
\(L(\frac12)\) \(\approx\) \(2.897404856\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-195. - 303. i)T \)
good3 \( 1 - 15.3iT - 81T^{2} \)
5 \( 1 - 41.6T + 625T^{2} \)
7 \( 1 - 62.4T + 2.40e3T^{2} \)
11 \( 1 + 122.T + 1.46e4T^{2} \)
13 \( 1 - 68.9iT - 2.85e4T^{2} \)
17 \( 1 - 297.T + 8.35e4T^{2} \)
23 \( 1 + 268.T + 2.79e5T^{2} \)
29 \( 1 + 561. iT - 7.07e5T^{2} \)
31 \( 1 - 252. iT - 9.23e5T^{2} \)
37 \( 1 - 2.40e3iT - 1.87e6T^{2} \)
41 \( 1 + 690. iT - 2.82e6T^{2} \)
43 \( 1 + 218.T + 3.41e6T^{2} \)
47 \( 1 - 83.1T + 4.87e6T^{2} \)
53 \( 1 - 4.38e3iT - 7.89e6T^{2} \)
59 \( 1 - 476. iT - 1.21e7T^{2} \)
61 \( 1 - 3.96e3T + 1.38e7T^{2} \)
67 \( 1 + 5.11e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.18e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.34e3T + 2.83e7T^{2} \)
79 \( 1 - 1.48e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.34e3T + 4.74e7T^{2} \)
89 \( 1 + 1.11e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.07e3iT - 8.85e7T^{2} \)
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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

−11.03786801493196115915247490576, −10.11879072064794033797741830586, −9.932121424225202504944471564479, −8.821399220683744538472275085822, −7.85281388924673466790589769053, −5.92820209908045225668875440640, −5.30536608875959321864972256618, −4.53316826443196583480778377211, −3.01286338146820285125510686770, −1.67647885150368378722580386044, 0.910949151912624280538921950588, 1.87076283074155666826148330651, 2.66338975567087718665377104622, 5.30056209995951942940705727535, 5.66275409038251639473264833197, 6.94417583258856271672699444100, 7.77738412209571714027960043035, 8.580215529393227472787806332455, 9.847844107131376943019493838058, 10.86966982705544697769761715985

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