Properties

Label 2-304-19.12-c4-0-20
Degree $2$
Conductor $304$
Sign $0.276 + 0.960i$
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  + (−10.1 + 5.84i)3-s + (20.8 + 36.0i)5-s − 24.1·7-s + (27.9 − 48.3i)9-s − 62.4·11-s + (−250. − 144. i)13-s + (−422. − 243. i)15-s + (−12.2 − 21.2i)17-s + (283. + 223. i)19-s + (245. − 141. i)21-s + (−302. + 524. i)23-s + (−555. + 962. i)25-s − 294. i·27-s + (249. + 144. i)29-s + 418. i·31-s + ⋯
L(s)  = 1  + (−1.12 + 0.649i)3-s + (0.833 + 1.44i)5-s − 0.493·7-s + (0.344 − 0.597i)9-s − 0.516·11-s + (−1.48 − 0.855i)13-s + (−1.87 − 1.08i)15-s + (−0.0424 − 0.0734i)17-s + (0.784 + 0.620i)19-s + (0.555 − 0.320i)21-s + (−0.572 + 0.991i)23-s + (−0.889 + 1.54i)25-s − 0.403i·27-s + (0.296 + 0.171i)29-s + 0.435i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.276 + 0.960i$
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: \((2,\ 304,\ (\ :2),\ 0.276 + 0.960i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1178550859\)
\(L(\frac12)\) \(\approx\) \(0.1178550859\)
\(L(3)\) 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 \( 1 + (-283. - 223. i)T \)
good3 \( 1 + (10.1 - 5.84i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-20.8 - 36.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 24.1T + 2.40e3T^{2} \)
11 \( 1 + 62.4T + 1.46e4T^{2} \)
13 \( 1 + (250. + 144. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (12.2 + 21.2i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (302. - 524. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-249. - 144. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 418. iT - 9.23e5T^{2} \)
37 \( 1 + 2.02e3iT - 1.87e6T^{2} \)
41 \( 1 + (717. - 414. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (632. + 1.09e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-421. + 729. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-3.14e3 - 1.81e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-4.62e3 + 2.67e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.47e3 + 2.54e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (4.19e3 + 2.42e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-4.78e3 + 2.76e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-792. - 1.37e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (4.62e3 - 2.67e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 740.T + 4.74e7T^{2} \)
89 \( 1 + (-4.78e3 - 2.76e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-2.96e3 + 1.71e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.65292960513025770586313838595, −10.10491683195928451893563300733, −9.642332922279736361832964190280, −7.69363029517324170725727346348, −6.80913567830100340691932649109, −5.72438243231703766585365449924, −5.21581724003418686284071927719, −3.47075112513971158063560439332, −2.35597281859385870486958997224, −0.04779616165092615242143350581, 1.01642307539522822164555685238, 2.35013717258802737058684676251, 4.63817141286302179023492004605, 5.28037034639370690270845721097, 6.25795287315246318580328119131, 7.15649981137548927662770349299, 8.487202003953696299988573792474, 9.540281286461445197181826050208, 10.17216237626645505469606782245, 11.70996487426125938196804748291

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