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Properties

Label 2-304-19.8-c4-0-37
Degree $2$
Conductor $304$
Sign $0.0317 + 0.999i$
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  + (10.7 + 6.21i)3-s + (20.5 − 35.6i)5-s − 44.5·7-s + (36.7 + 63.6i)9-s − 201.·11-s + (−5.67 + 3.27i)13-s + (443. − 255. i)15-s + (274. − 476. i)17-s + (200. − 300. i)19-s + (−479. − 276. i)21-s + (−75.0 − 130. i)23-s + (−534. − 926. i)25-s − 93.7i·27-s + (236. − 136. i)29-s + 1.73e3i·31-s + ⋯
L(s)  = 1  + (1.19 + 0.690i)3-s + (0.823 − 1.42i)5-s − 0.909·7-s + (0.453 + 0.785i)9-s − 1.66·11-s + (−0.0335 + 0.0193i)13-s + (1.96 − 1.13i)15-s + (0.950 − 1.64i)17-s + (0.556 − 0.831i)19-s + (−1.08 − 0.627i)21-s + (−0.141 − 0.245i)23-s + (−0.855 − 1.48i)25-s − 0.128i·27-s + (0.280 − 0.162i)29-s + 1.80i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.0317 + 0.999i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.0317 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.371234291\)
\(L(\frac12)\) \(\approx\) \(2.371234291\)
\(L(3)\) 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 \( 1 + (-200. + 300. i)T \)
good3 \( 1 + (-10.7 - 6.21i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-20.5 + 35.6i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + 44.5T + 2.40e3T^{2} \)
11 \( 1 + 201.T + 1.46e4T^{2} \)
13 \( 1 + (5.67 - 3.27i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-274. + 476. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (75.0 + 130. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-236. + 136. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 1.73e3iT - 9.23e5T^{2} \)
37 \( 1 + 280. iT - 1.87e6T^{2} \)
41 \( 1 + (1.51e3 + 877. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.24e3 + 2.15e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.64e3 + 2.84e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (657. - 379. i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-1.62e3 - 940. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.59e3 - 4.50e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-2.72e3 + 1.57e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (1.05e3 + 611. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (458. - 794. i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-6.24e3 - 3.60e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 7.16e3T + 4.74e7T^{2} \)
89 \( 1 + (1.01e3 - 588. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-4.90e3 - 2.83e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−10.26449748233814072998966611891, −9.800585235122020397525184411787, −9.018879750829302370279933781407, −8.367345032708017433963937133241, −7.16065421338275724234724641157, −5.40593426253908148729473759780, −4.90411658649034579899902149457, −3.29196555899353030789654745047, −2.42894305226936678073065549532, −0.57324115891088953423197788240, 1.83109135113842410456873032931, 2.82736717765009025466472721904, 3.42381707325954277709795738596, 5.72283852338890815054948059477, 6.47100147467255217542750842071, 7.68283289075529194113499221133, 8.084943306766177698766977356078, 9.751232891785306204161694434046, 10.04677240860807486511683611921, 11.08488595922991614447574396732

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