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Properties

Label 2-304-19.12-c4-0-19
Degree $2$
Conductor $304$
Sign $0.998 - 0.0604i$
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  + (8.70 − 5.02i)3-s + (4.61 + 8.00i)5-s − 0.274·7-s + (10.0 − 17.4i)9-s − 45.5·11-s + (111. + 64.2i)13-s + (80.4 + 46.4i)15-s + (212. + 367. i)17-s + (280. − 227. i)19-s + (−2.39 + 1.38i)21-s + (286. − 496. i)23-s + (269. − 467. i)25-s + 612. i·27-s + (763. + 440. i)29-s + 1.53e3i·31-s + ⋯
L(s)  = 1  + (0.967 − 0.558i)3-s + (0.184 + 0.320i)5-s − 0.00560·7-s + (0.124 − 0.215i)9-s − 0.376·11-s + (0.658 + 0.380i)13-s + (0.357 + 0.206i)15-s + (0.733 + 1.27i)17-s + (0.776 − 0.630i)19-s + (−0.00542 + 0.00312i)21-s + (0.542 − 0.939i)23-s + (0.431 − 0.747i)25-s + 0.839i·27-s + (0.908 + 0.524i)29-s + 1.59i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0604i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 - 0.0604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.998 - 0.0604i$
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: \((2,\ 304,\ (\ :2),\ 0.998 - 0.0604i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.023099738\)
\(L(\frac12)\) \(\approx\) \(3.023099738\)
\(L(3)\) 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 \( 1 + (-280. + 227. i)T \)
good3 \( 1 + (-8.70 + 5.02i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-4.61 - 8.00i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 0.274T + 2.40e3T^{2} \)
11 \( 1 + 45.5T + 1.46e4T^{2} \)
13 \( 1 + (-111. - 64.2i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-212. - 367. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-286. + 496. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-763. - 440. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.53e3iT - 9.23e5T^{2} \)
37 \( 1 - 71.9iT - 1.87e6T^{2} \)
41 \( 1 + (-1.11e3 + 642. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (603. + 1.04e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.34e3 + 2.33e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.83e3 - 1.05e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.85e3 - 2.80e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.24e3 + 2.14e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-656. - 379. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-5.26e3 + 3.04e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-3.51e3 - 6.08e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.79e3 + 1.61e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 5.58e3T + 4.74e7T^{2} \)
89 \( 1 + (4.91e3 + 2.83e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.02e4 + 5.93e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.89979833618750651541562213902, −10.24385883020564998234579117503, −8.845049047647259665607798441668, −8.383372801295567992089976212036, −7.27651095087601921600752489510, −6.39641146522550598186970441225, −5.03659156336455411635932133720, −3.47823099026802766895734543858, −2.52792652574807399480255751993, −1.23359569097804600565980127543, 0.996015702623362526233706648427, 2.75410648656938165107961162546, 3.59123506609877797944972835700, 4.93203095533801575077269257538, 5.97178268035818837757070150100, 7.54466215502892012807758793310, 8.231754183699395367506511707563, 9.497394954114101454136882652210, 9.630916237877171794922079893778, 11.02082986530316396624029255031

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