Properties

Label 2-304-19.7-c5-0-15
Degree $2$
Conductor $304$
Sign $0.682 - 0.730i$
Analytic cond. $48.7566$
Root an. cond. $6.98259$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.4 − 18.1i)3-s + (50.5 + 87.6i)5-s − 95.5·7-s + (−97.0 + 168. i)9-s + 119.·11-s + (297. − 515. i)13-s + (1.05e3 − 1.83e3i)15-s + (−459. − 796. i)17-s + (806. − 1.35e3i)19-s + (998. + 1.73e3i)21-s + (−2.12e3 + 3.67e3i)23-s + (−3.55e3 + 6.16e3i)25-s − 1.02e3·27-s + (−2.21e3 + 3.83e3i)29-s + 4.95e3·31-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)3-s + (0.905 + 1.56i)5-s − 0.736·7-s + (−0.399 + 0.691i)9-s + 0.297·11-s + (0.488 − 0.845i)13-s + (1.21 − 2.10i)15-s + (−0.385 − 0.668i)17-s + (0.512 − 0.858i)19-s + (0.494 + 0.856i)21-s + (−0.837 + 1.45i)23-s + (−1.13 + 1.97i)25-s − 0.269·27-s + (−0.489 + 0.846i)29-s + 0.926·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(48.7566\)
Root analytic conductor: \(6.98259\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :5/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.403466434\)
\(L(\frac12)\) \(\approx\) \(1.403466434\)
\(L(\frac{7}{2})\) 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 \( 1 + (-806. + 1.35e3i)T \)
good3 \( 1 + (10.4 + 18.1i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-50.5 - 87.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + 95.5T + 1.68e4T^{2} \)
11 \( 1 - 119.T + 1.61e5T^{2} \)
13 \( 1 + (-297. + 515. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (459. + 796. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (2.12e3 - 3.67e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (2.21e3 - 3.83e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 4.95e3T + 2.86e7T^{2} \)
37 \( 1 - 7.65e3T + 6.93e7T^{2} \)
41 \( 1 + (-2.00e3 - 3.47e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-5.22e3 - 9.05e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (4.31e3 - 7.47e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.85e4 + 3.21e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.07e4 - 3.59e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.24e3 - 3.88e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.84e4 - 3.20e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (2.34e4 + 4.05e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-4.06e4 - 7.04e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (6.65e3 + 1.15e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.83e4T + 3.93e9T^{2} \)
89 \( 1 + (7.30e3 - 1.26e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-6.18e4 - 1.07e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−11.20463445291894596664133974022, −10.11894923212751749202805176986, −9.383621844226628457047362777405, −7.65384717290220458748175104003, −6.91845295494109020489655133499, −6.26277986834241357933694462013, −5.55545660899940821042889843380, −3.37061099047214485127764001507, −2.41112973718335195586745280283, −1.04526766037371524022363179548, 0.47108076951842243861612623743, 1.91469918188903005754254687799, 3.99307294114904889384057545814, 4.58082225326975391892274887259, 5.78327090527878743117458756377, 6.27660421702092515377230838229, 8.268861496383506546630311810822, 9.176683769751858342429117813124, 9.780532911620182625646809812267, 10.48083733058950705589366421293

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