Properties

Label 2-304-19.18-c6-0-7
Degree $2$
Conductor $304$
Sign $-0.364 - 0.931i$
Analytic cond. $69.9364$
Root an. cond. $8.36280$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.8i·3-s − 18.1·5-s − 85.5·7-s + 536.·9-s + 316.·11-s + 2.45e3i·13-s + 251. i·15-s − 3.22e3·17-s + (−2.50e3 − 6.38e3i)19-s + 1.18e3i·21-s + 7.82e3·23-s − 1.52e4·25-s − 1.75e4i·27-s + 1.89e4i·29-s − 3.23e4i·31-s + ⋯
L(s)  = 1  − 0.514i·3-s − 0.144·5-s − 0.249·7-s + 0.735·9-s + 0.237·11-s + 1.11i·13-s + 0.0744i·15-s − 0.656·17-s + (−0.364 − 0.931i)19-s + 0.128i·21-s + 0.643·23-s − 0.978·25-s − 0.892i·27-s + 0.778i·29-s − 1.08i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.364 - 0.931i$
Analytic conductor: \(69.9364\)
Root analytic conductor: \(8.36280\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :3),\ -0.364 - 0.931i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7698568172\)
\(L(\frac12)\) \(\approx\) \(0.7698568172\)
\(L(4)\) 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 \( 1 + (2.50e3 + 6.38e3i)T \)
good3 \( 1 + 13.8iT - 729T^{2} \)
5 \( 1 + 18.1T + 1.56e4T^{2} \)
7 \( 1 + 85.5T + 1.17e5T^{2} \)
11 \( 1 - 316.T + 1.77e6T^{2} \)
13 \( 1 - 2.45e3iT - 4.82e6T^{2} \)
17 \( 1 + 3.22e3T + 2.41e7T^{2} \)
23 \( 1 - 7.82e3T + 1.48e8T^{2} \)
29 \( 1 - 1.89e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.23e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.54e4iT - 2.56e9T^{2} \)
41 \( 1 - 9.41e4iT - 4.75e9T^{2} \)
43 \( 1 + 3.92e4T + 6.32e9T^{2} \)
47 \( 1 + 9.23e4T + 1.07e10T^{2} \)
53 \( 1 + 1.30e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.80e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.24e5T + 5.15e10T^{2} \)
67 \( 1 - 1.54e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.77e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.14e5T + 1.51e11T^{2} \)
79 \( 1 - 4.62e5iT - 2.43e11T^{2} \)
83 \( 1 + 6.01e5T + 3.26e11T^{2} \)
89 \( 1 - 2.75e5iT - 4.96e11T^{2} \)
97 \( 1 - 8.52e5iT - 8.32e11T^{2} \)
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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

−11.20789681387645353942995319983, −9.903919437554543194662569715764, −9.152944354596763731500465601457, −8.048888388096314714074956046305, −6.91284655650471147881472262047, −6.46473958335586251845372492383, −4.84678054335817463343776342359, −3.90273605606814934935855696292, −2.36878492094067651821222798700, −1.26166886921367116336565412940, 0.18635819042941145036396823766, 1.70362945589708948622108248513, 3.26065901609252338395042729524, 4.19406608367654231147673758780, 5.31025323373162772075846394155, 6.46200800273943245568129646050, 7.54309919773278396623256320267, 8.534970599339238091278448809058, 9.634033967804587403795555155003, 10.31686936678916897818725449616

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