Properties

Label 2-304-19.12-c4-0-7
Degree $2$
Conductor $304$
Sign $-0.983 - 0.180i$
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  + (−3.54 + 2.04i)3-s + (15.5 + 27.0i)5-s + 2.67·7-s + (−32.1 + 55.6i)9-s − 42.3·11-s + (113. + 65.6i)13-s + (−110. − 63.8i)15-s + (143. + 247. i)17-s + (−326. + 153. i)19-s + (−9.50 + 5.48i)21-s + (158. − 273. i)23-s + (−173. + 300. i)25-s − 594. i·27-s + (−188. − 108. i)29-s + 475. i·31-s + ⋯
L(s)  = 1  + (−0.394 + 0.227i)3-s + (0.623 + 1.08i)5-s + 0.0546·7-s + (−0.396 + 0.686i)9-s − 0.349·11-s + (0.673 + 0.388i)13-s + (−0.491 − 0.283i)15-s + (0.495 + 0.858i)17-s + (−0.904 + 0.426i)19-s + (−0.0215 + 0.0124i)21-s + (0.299 − 0.517i)23-s + (−0.277 + 0.480i)25-s − 0.815i·27-s + (−0.224 − 0.129i)29-s + 0.495i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.180i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.983 - 0.180i$
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: \((2,\ 304,\ (\ :2),\ -0.983 - 0.180i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.116638665\)
\(L(\frac12)\) \(\approx\) \(1.116638665\)
\(L(3)\) 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 \( 1 + (326. - 153. i)T \)
good3 \( 1 + (3.54 - 2.04i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-15.5 - 27.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 2.67T + 2.40e3T^{2} \)
11 \( 1 + 42.3T + 1.46e4T^{2} \)
13 \( 1 + (-113. - 65.6i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-143. - 247. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-158. + 273. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (188. + 108. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 475. iT - 9.23e5T^{2} \)
37 \( 1 + 1.74e3iT - 1.87e6T^{2} \)
41 \( 1 + (104. - 60.5i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.23e3 - 2.14e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (168. - 292. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (3.65e3 + 2.11e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.91e3 - 2.83e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.94e3 - 3.37e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (2.80e3 + 1.61e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (4.05e3 - 2.33e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-803. - 1.39e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.90e3 + 1.68e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 6.54e3T + 4.74e7T^{2} \)
89 \( 1 + (-3.17e3 - 1.83e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-4.87e3 + 2.81e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.05126545364598111929677707895, −10.77764826365338097968837690517, −9.942762360755624035500832005756, −8.661426248706559080730731060120, −7.66103625189756977199498562070, −6.36627858976013122505063785888, −5.81982343425238140994521647201, −4.44799275395908739925763318957, −3.02363449413613349988805208936, −1.82981190950712868585995716263, 0.35548331955496102214424331738, 1.48063275133485385875827019359, 3.16570066797020616339116297322, 4.75146814378429760538272573946, 5.61792564380675130724880022029, 6.48142664339527121593735452502, 7.83346363831733650062066273070, 8.896948692621825719613462672718, 9.495952648036000180906780959783, 10.72423265815214161133344010745

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