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Properties

Label 2-304-19.12-c4-0-10
Degree $2$
Conductor $304$
Sign $0.292 - 0.956i$
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  + (5.59 − 3.22i)3-s + (−4.68 − 8.12i)5-s − 33.5·7-s + (−19.6 + 34.0i)9-s + 58.1·11-s + (−29.5 − 17.0i)13-s + (−52.4 − 30.2i)15-s + (−9.99 − 17.3i)17-s + (115. + 342. i)19-s + (−187. + 108. i)21-s + (−459. + 796. i)23-s + (268. − 465. i)25-s + 776. i·27-s + (685. + 395. i)29-s + 517. i·31-s + ⋯
L(s)  = 1  + (0.621 − 0.358i)3-s + (−0.187 − 0.324i)5-s − 0.684·7-s + (−0.242 + 0.420i)9-s + 0.480·11-s + (−0.175 − 0.101i)13-s + (−0.233 − 0.134i)15-s + (−0.0345 − 0.0598i)17-s + (0.318 + 0.947i)19-s + (−0.425 + 0.245i)21-s + (−0.869 + 1.50i)23-s + (0.429 − 0.744i)25-s + 1.06i·27-s + (0.815 + 0.470i)29-s + 0.538i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.292 - 0.956i$
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: \((2,\ 304,\ (\ :2),\ 0.292 - 0.956i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.528703891\)
\(L(\frac12)\) \(\approx\) \(1.528703891\)
\(L(3)\) 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 \( 1 + (-115. - 342. i)T \)
good3 \( 1 + (-5.59 + 3.22i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (4.68 + 8.12i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + 33.5T + 2.40e3T^{2} \)
11 \( 1 - 58.1T + 1.46e4T^{2} \)
13 \( 1 + (29.5 + 17.0i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (9.99 + 17.3i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (459. - 796. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-685. - 395. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 517. iT - 9.23e5T^{2} \)
37 \( 1 + 181. iT - 1.87e6T^{2} \)
41 \( 1 + (-738. + 426. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.24e3 - 2.14e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (158. - 274. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.24e3 + 719. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (751. - 433. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (403. - 699. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-5.97e3 - 3.44e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (617. - 356. i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-3.60e3 - 6.23e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (3.64e3 - 2.10e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 2.98e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.56e3 - 902. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (599. - 346. i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.37414844036890360083823951251, −10.17076445215107925087771352780, −9.327330873277564537391188525781, −8.334866301224955429099044633555, −7.59642787582826411352332706233, −6.44872547194185505001971985564, −5.29583533610832251651523320137, −3.87613252728632722252056077897, −2.77225530500086256830378165732, −1.35387635038162836399759665815, 0.44432563401834314119089551337, 2.50625902837404989539352689362, 3.46449888577719398242957941971, 4.52268056103582330128903289926, 6.12930208165842764679956342368, 6.90599704469781855977216431801, 8.174450723903694945697081673345, 9.117871088899010109477503625235, 9.763050278277329614789951179236, 10.81157042699694225766405678516

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