Properties

Label 2-304-19.12-c4-0-18
Degree $2$
Conductor $304$
Sign $0.529 + 0.848i$
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  + (−1.30 + 0.754i)3-s + (1.35 + 2.34i)5-s − 79.4·7-s + (−39.3 + 68.1i)9-s − 109.·11-s + (269. + 155. i)13-s + (−3.54 − 2.04i)15-s + (−212. − 367. i)17-s + (333. + 137. i)19-s + (103. − 59.9i)21-s + (132. − 228. i)23-s + (308. − 534. i)25-s − 241. i·27-s + (−203. − 117. i)29-s − 648. i·31-s + ⋯
L(s)  = 1  + (−0.145 + 0.0838i)3-s + (0.0541 + 0.0938i)5-s − 1.62·7-s + (−0.485 + 0.841i)9-s − 0.901·11-s + (1.59 + 0.922i)13-s + (−0.0157 − 0.00908i)15-s + (−0.735 − 1.27i)17-s + (0.924 + 0.381i)19-s + (0.235 − 0.136i)21-s + (0.249 − 0.432i)23-s + (0.494 − 0.855i)25-s − 0.330i·27-s + (−0.241 − 0.139i)29-s − 0.674i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.529 + 0.848i$
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: \((2,\ 304,\ (\ :2),\ 0.529 + 0.848i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9542009548\)
\(L(\frac12)\) \(\approx\) \(0.9542009548\)
\(L(3)\) 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 \( 1 + (-333. - 137. i)T \)
good3 \( 1 + (1.30 - 0.754i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-1.35 - 2.34i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 79.4T + 2.40e3T^{2} \)
11 \( 1 + 109.T + 1.46e4T^{2} \)
13 \( 1 + (-269. - 155. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (212. + 367. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-132. + 228. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (203. + 117. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 648. iT - 9.23e5T^{2} \)
37 \( 1 - 626. iT - 1.87e6T^{2} \)
41 \( 1 + (415. - 239. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-622. - 1.07e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-822. + 1.42e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (4.11e3 + 2.37e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-4.22e3 + 2.44e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-3.54e3 + 6.14e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (51.8 + 29.9i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (5.41e3 - 3.12e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (1.20e3 + 2.09e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-3.68e3 + 2.12e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 6.64e3T + 4.74e7T^{2} \)
89 \( 1 + (-8.48e3 - 4.90e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (4.06e3 - 2.34e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.94995439289575875291607912393, −9.980753324715721294446882714946, −9.145868181829470540213123746858, −8.115417412034160880880203174436, −6.83322809181535740532363313039, −6.09197727152444486340851274515, −4.90830915550173255058422571955, −3.46903950687616469794003900288, −2.43101978279102668355681353892, −0.38032931172260670026222884738, 0.931278156531214412829401313490, 3.01262876059066241007472553404, 3.66566187578184263090529118647, 5.58151945928307612711972593276, 6.16850822391280933347034723618, 7.21557170170894778115428276726, 8.597717085135616488833272434418, 9.248141368421305207290675605219, 10.39379079812363933148942755970, 11.06541500135968288264948563857

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