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Properties

Label 2-304-19.12-c4-0-33
Degree $2$
Conductor $304$
Sign $-0.829 + 0.558i$
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  + (5.28 − 3.05i)3-s + (1.28 + 2.22i)5-s + 6.16·7-s + (−21.8 + 37.9i)9-s − 70.7·11-s + (−53.8 − 31.1i)13-s + (13.5 + 7.83i)15-s + (−222. − 385. i)17-s + (126. − 338. i)19-s + (32.5 − 18.8i)21-s + (−139. + 242. i)23-s + (309. − 535. i)25-s + 761. i·27-s + (−1.22e3 − 709. i)29-s + 386. i·31-s + ⋯
L(s)  = 1  + (0.587 − 0.338i)3-s + (0.0513 + 0.0889i)5-s + 0.125·7-s + (−0.270 + 0.468i)9-s − 0.584·11-s + (−0.318 − 0.184i)13-s + (0.0603 + 0.0348i)15-s + (−0.769 − 1.33i)17-s + (0.349 − 0.937i)19-s + (0.0738 − 0.0426i)21-s + (−0.264 + 0.458i)23-s + (0.494 − 0.856i)25-s + 1.04i·27-s + (−1.46 − 0.843i)29-s + 0.402i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.829 + 0.558i$
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: \((2,\ 304,\ (\ :2),\ -0.829 + 0.558i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9390571156\)
\(L(\frac12)\) \(\approx\) \(0.9390571156\)
\(L(3)\) 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 \( 1 + (-126. + 338. i)T \)
good3 \( 1 + (-5.28 + 3.05i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-1.28 - 2.22i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 6.16T + 2.40e3T^{2} \)
11 \( 1 + 70.7T + 1.46e4T^{2} \)
13 \( 1 + (53.8 + 31.1i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (222. + 385. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (139. - 242. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.22e3 + 709. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 386. iT - 9.23e5T^{2} \)
37 \( 1 + 2.28e3iT - 1.87e6T^{2} \)
41 \( 1 + (438. - 252. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-405. - 702. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-445. + 772. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-537. - 310. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (1.30e3 - 751. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.48e3 - 2.56e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.38e3 + 1.95e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-2.77e3 + 1.59e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (223. + 386. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-3.06e3 + 1.76e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.12e4T + 4.74e7T^{2} \)
89 \( 1 + (2.94e3 + 1.69e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (7.58e3 - 4.37e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.82734022519231968394427084262, −9.599989502933910147580366590077, −8.792012578210138971389425504206, −7.71170740632073888281387819516, −7.10138405838499152218495360993, −5.60974444678098528549868259464, −4.61983302543986538004458248910, −2.96806719237398367640530829502, −2.13350995025559946805111210422, −0.24388136282937668785192851900, 1.71871665025370195383007885914, 3.13042368842652872937884707755, 4.15498136617484778644450998472, 5.45704571225372416732491162362, 6.55779875697946599862403563814, 7.85162309868323737054177598906, 8.640449218941084645609845740737, 9.519175727202488064469273580335, 10.42970819831817788590242939575, 11.37859289497265652835385008871

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