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Properties

Label 2-304-19.8-c4-0-4
Degree $2$
Conductor $304$
Sign $-0.448 - 0.893i$
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  + (−0.620 − 0.358i)3-s + (16.9 − 29.3i)5-s − 38.8·7-s + (−40.2 − 69.7i)9-s − 37.5·11-s + (−175. + 101. i)13-s + (−21.0 + 12.1i)15-s + (−1.83 + 3.17i)17-s + (−56.3 + 356. i)19-s + (24.0 + 13.9i)21-s + (211. + 365. i)23-s + (−261. − 452. i)25-s + 115. i·27-s + (385. − 222. i)29-s + 773. i·31-s + ⋯
L(s)  = 1  + (−0.0689 − 0.0398i)3-s + (0.677 − 1.17i)5-s − 0.792·7-s + (−0.496 − 0.860i)9-s − 0.310·11-s + (−1.03 + 0.600i)13-s + (−0.0934 + 0.0539i)15-s + (−0.00633 + 0.0109i)17-s + (−0.156 + 0.987i)19-s + (0.0546 + 0.0315i)21-s + (0.398 + 0.691i)23-s + (−0.417 − 0.723i)25-s + 0.158i·27-s + (0.457 − 0.264i)29-s + 0.804i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.448 - 0.893i$
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: \((2,\ 304,\ (\ :2),\ -0.448 - 0.893i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3026492985\)
\(L(\frac12)\) \(\approx\) \(0.3026492985\)
\(L(3)\) 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 \( 1 + (56.3 - 356. i)T \)
good3 \( 1 + (0.620 + 0.358i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-16.9 + 29.3i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + 38.8T + 2.40e3T^{2} \)
11 \( 1 + 37.5T + 1.46e4T^{2} \)
13 \( 1 + (175. - 101. i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (-211. - 365. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-385. + 222. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 773. iT - 9.23e5T^{2} \)
37 \( 1 + 1.46e3iT - 1.87e6T^{2} \)
41 \( 1 + (-2.32e3 - 1.34e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (116. - 201. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-925. - 1.60e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (1.86e3 - 1.07e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-879. - 507. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (301. + 522. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (5.99e3 - 3.46e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (6.25e3 + 3.60e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (3.53e3 - 6.11e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (5.21e3 + 3.01e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 1.06e3T + 4.74e7T^{2} \)
89 \( 1 + (1.61e3 - 933. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (1.21e3 + 699. i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−11.59065113061419186295932890050, −10.17120838796298323445573170653, −9.394465175447069242870550722525, −8.866722223941774976011310205214, −7.53236540075551124683763323321, −6.28374908150217491413812735139, −5.50640591752656875487110825685, −4.32915845724770433032598631773, −2.87347430641499476748924750347, −1.32923021701367671254456980175, 0.091512028941647537252826137027, 2.44251104078347021800384615771, 2.94795838825335356054700784119, 4.79747091962540678120936927313, 5.89814013202103433529087690869, 6.81814399920074802777658039865, 7.70578217757216975573742789438, 9.024800718463205178941972607607, 10.13923152704190395665382783366, 10.53505279260851420001841944178

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