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Properties

Label 2-304-19.18-c2-0-18
Degree $2$
Conductor $304$
Sign $-0.526 - 0.850i$
Analytic cond. $8.28340$
Root an. cond. $2.87808$
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  − 5.38i·3-s − 4·5-s + 7-s − 19.9·9-s − 14·11-s + 16.1i·13-s + 21.5i·15-s + 23·17-s + (−10 − 16.1i)19-s − 5.38i·21-s + 23-s − 9·25-s + 59.2i·27-s − 48.4i·29-s + 32.3i·31-s + ⋯
L(s)  = 1  − 1.79i·3-s − 0.800·5-s + 0.142·7-s − 2.22·9-s − 1.27·11-s + 1.24i·13-s + 1.43i·15-s + 1.35·17-s + (−0.526 − 0.850i)19-s − 0.256i·21-s + 0.0434·23-s − 0.359·25-s + 2.19i·27-s − 1.67i·29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(8.28340\)
Root analytic conductor: \(2.87808\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1),\ -0.526 - 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.140609 + 0.252400i\)
\(L(\frac12)\) \(\approx\) \(0.140609 + 0.252400i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (10 + 16.1i)T \)
good3 \( 1 + 5.38iT - 9T^{2} \)
5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - T + 49T^{2} \)
11 \( 1 + 14T + 121T^{2} \)
13 \( 1 - 16.1iT - 169T^{2} \)
17 \( 1 - 23T + 289T^{2} \)
23 \( 1 - T + 529T^{2} \)
29 \( 1 + 48.4iT - 841T^{2} \)
31 \( 1 - 32.3iT - 961T^{2} \)
37 \( 1 - 32.3iT - 1.36e3T^{2} \)
41 \( 1 - 32.3iT - 1.68e3T^{2} \)
43 \( 1 + 68T + 1.84e3T^{2} \)
47 \( 1 + 26T + 2.20e3T^{2} \)
53 \( 1 + 80.7iT - 2.80e3T^{2} \)
59 \( 1 + 16.1iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 - 16.1iT - 4.48e3T^{2} \)
71 \( 1 - 32.3iT - 5.04e3T^{2} \)
73 \( 1 + 7T + 5.32e3T^{2} \)
79 \( 1 + 96.9iT - 6.24e3T^{2} \)
83 \( 1 + 32T + 6.88e3T^{2} \)
89 \( 1 + 129. iT - 7.92e3T^{2} \)
97 \( 1 + 96.9iT - 9.40e3T^{2} \)
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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

−11.51089124983979288171443922154, −9.976499739868240122773070602933, −8.429720045638069011878038389143, −7.933928393804037235717706352640, −7.10142279471633627440903028542, −6.21014687627168187753861127816, −4.87069850791155669445467921996, −3.08151311742406004573393876917, −1.77142759476623953963895995082, −0.13452042665839393995238579523, 3.05436377462331923120413964721, 3.82771571257552130562890958955, 5.07188702153644665249628450593, 5.67758016848150475117776113762, 7.76197182528621233414732282366, 8.268795072332655496933284078998, 9.529168209752329272359064676833, 10.45755220129903749774002090404, 10.74081752123605529282444153474, 11.92291021531373761057622766743

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