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Properties

Label 2-304-19.12-c4-0-14
Degree $2$
Conductor $304$
Sign $0.969 - 0.243i$
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  + (0.962 − 0.555i)3-s + (−13.4 − 23.2i)5-s + 62.8·7-s + (−39.8 + 69.0i)9-s − 134.·11-s + (222. + 128. i)13-s + (−25.8 − 14.9i)15-s + (52.8 + 91.5i)17-s + (−233. + 275. i)19-s + (60.4 − 34.9i)21-s + (184. − 319. i)23-s + (−47.3 + 81.9i)25-s + 178. i·27-s + (1.00e3 + 581. i)29-s − 1.29e3i·31-s + ⋯
L(s)  = 1  + (0.106 − 0.0617i)3-s + (−0.536 − 0.929i)5-s + 1.28·7-s + (−0.492 + 0.852i)9-s − 1.11·11-s + (1.31 + 0.759i)13-s + (−0.114 − 0.0662i)15-s + (0.182 + 0.316i)17-s + (−0.646 + 0.762i)19-s + (0.137 − 0.0791i)21-s + (0.348 − 0.603i)23-s + (−0.0757 + 0.131i)25-s + 0.245i·27-s + (1.19 + 0.691i)29-s − 1.35i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.983757281\)
\(L(\frac12)\) \(\approx\) \(1.983757281\)
\(L(3)\) 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 \( 1 + (233. - 275. i)T \)
good3 \( 1 + (-0.962 + 0.555i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (13.4 + 23.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 62.8T + 2.40e3T^{2} \)
11 \( 1 + 134.T + 1.46e4T^{2} \)
13 \( 1 + (-222. - 128. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-52.8 - 91.5i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-184. + 319. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.00e3 - 581. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.29e3iT - 9.23e5T^{2} \)
37 \( 1 - 935. iT - 1.87e6T^{2} \)
41 \( 1 + (-1.65e3 + 953. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.34e3 - 2.32e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.15e3 + 2.00e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-3.26e3 - 1.88e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-1.39e3 + 802. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (3.29e3 - 5.70e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (883. + 510. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.05e3 + 4.07e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-1.21e3 - 2.10e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (8.53e3 - 4.92e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 6.63e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.01e4 - 5.85e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.97e3 + 1.13e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.06696254463987258974313670350, −10.49154349022341664648305535015, −8.749727782436417229684867238184, −8.330589264658408764493949359775, −7.66894377328065004561660166042, −5.98021665321111720977356002481, −4.92216930733172941425137773220, −4.16715498056037473265278562318, −2.33110833481059623208432247838, −1.04978868420444380769684264907, 0.76434339298912796975325149958, 2.61906184803421866337056663062, 3.60814878766413324933964869067, 4.98489613034729563278528439265, 6.10611218442108090953814450314, 7.31405809256388298575570612036, 8.156919109554925260455742443721, 8.926157918893090463625510244692, 10.50229351903642944801139582919, 10.97345752111797085115013040295

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