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Properties

Label 2-304-76.7-c4-0-36
Degree $2$
Conductor $304$
Sign $-0.500 + 0.865i$
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  + (14.4 − 8.36i)3-s + (−8.09 − 14.0i)5-s + 22.9i·7-s + (99.4 − 172. i)9-s − 83.7i·11-s + (6.17 − 10.6i)13-s + (−234. − 135. i)15-s + (69.3 + 120. i)17-s + (−242. − 267. i)19-s + (191. + 331. i)21-s + (−123. − 71.1i)23-s + (181. − 314. i)25-s − 1.97e3i·27-s + (272. − 471. i)29-s + 48.9i·31-s + ⋯
L(s)  = 1  + (1.60 − 0.929i)3-s + (−0.323 − 0.561i)5-s + 0.467i·7-s + (1.22 − 2.12i)9-s − 0.692i·11-s + (0.0365 − 0.0632i)13-s + (−1.04 − 0.602i)15-s + (0.239 + 0.415i)17-s + (−0.671 − 0.741i)19-s + (0.434 + 0.752i)21-s + (−0.233 − 0.134i)23-s + (0.290 − 0.502i)25-s − 2.70i·27-s + (0.323 − 0.560i)29-s + 0.0509i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.500 + 0.865i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.500 + 0.865i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.057286270\)
\(L(\frac12)\) \(\approx\) \(3.057286270\)
\(L(3)\) 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 \( 1 + (242. + 267. i)T \)
good3 \( 1 + (-14.4 + 8.36i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (8.09 + 14.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 22.9iT - 2.40e3T^{2} \)
11 \( 1 + 83.7iT - 1.46e4T^{2} \)
13 \( 1 + (-6.17 + 10.6i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-69.3 - 120. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (123. + 71.1i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-272. + 471. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 48.9iT - 9.23e5T^{2} \)
37 \( 1 + 1.10e3T + 1.87e6T^{2} \)
41 \( 1 + (928. + 1.60e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-315. + 182. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.94e3 - 1.12e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (2.62e3 - 4.55e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (3.25e3 - 1.88e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (143. - 248. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.75e3 - 1.01e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-6.22e3 + 3.59e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (1.50e3 + 2.60e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-5.66e3 + 3.27e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.11e4iT - 4.74e7T^{2} \)
89 \( 1 + (893. - 1.54e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (6.15e3 + 1.06e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−10.72820040207330706770398649965, −9.357417775544483812614785139357, −8.637246593418346993183821982315, −8.166718783287940050163427282841, −7.11927711384252259551580798905, −6.01961226833721704244114933213, −4.31403726426260308948877880495, −3.13697858646369412921097601760, −2.10392413442819080432909978243, −0.75202903132627164968120628797, 1.91479922177782602204175082191, 3.16324239564289452545199671723, 3.93060249487276530670706825702, 4.96596040809816804466743102978, 6.88508440829956199133389371084, 7.73881331302873354334764061578, 8.562828919626400753958829909232, 9.586639040039518021920073074293, 10.23311618815728324376875202773, 11.02956879942326883640212451075

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