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Properties

Label 2-304-76.7-c4-0-0
Degree $2$
Conductor $304$
Sign $-0.374 + 0.927i$
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  + (−11.6 + 6.70i)3-s + (−20.2 − 35.0i)5-s + 70.8i·7-s + (49.4 − 85.6i)9-s + 179. i·11-s + (−14.7 + 25.6i)13-s + (470. + 271. i)15-s + (241. + 417. i)17-s + (−277. − 231. i)19-s + (−475. − 823. i)21-s + (71.3 + 41.1i)23-s + (−507. + 879. i)25-s + 240. i·27-s + (−476. + 824. i)29-s + 934. i·31-s + ⋯
L(s)  = 1  + (−1.29 + 0.745i)3-s + (−0.810 − 1.40i)5-s + 1.44i·7-s + (0.610 − 1.05i)9-s + 1.47i·11-s + (−0.0874 + 0.151i)13-s + (2.09 + 1.20i)15-s + (0.834 + 1.44i)17-s + (−0.768 − 0.640i)19-s + (−1.07 − 1.86i)21-s + (0.134 + 0.0778i)23-s + (−0.812 + 1.40i)25-s + 0.330i·27-s + (−0.566 + 0.980i)29-s + 0.972i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ -0.374 + 0.927i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.08493177972\)
\(L(\frac12)\) \(\approx\) \(0.08493177972\)
\(L(3)\) 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 \( 1 + (277. + 231. i)T \)
good3 \( 1 + (11.6 - 6.70i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (20.2 + 35.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 70.8iT - 2.40e3T^{2} \)
11 \( 1 - 179. iT - 1.46e4T^{2} \)
13 \( 1 + (14.7 - 25.6i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-241. - 417. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-71.3 - 41.1i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (476. - 824. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 934. iT - 9.23e5T^{2} \)
37 \( 1 - 71.7T + 1.87e6T^{2} \)
41 \( 1 + (303. + 525. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-196. + 113. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.91e3 + 1.68e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.15e3 + 3.73e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (4.16e3 - 2.40e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.71e3 + 4.70e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (561. + 324. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.84e3 + 4.52e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (1.76e3 + 3.05e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (2.79e3 - 1.61e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 5.84e3iT - 4.74e7T^{2} \)
89 \( 1 + (4.16e3 - 7.22e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (4.62e3 + 8.01e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−11.91982505024386405477368369374, −10.92552387599234019624676110068, −9.851973728346655732170491260019, −8.965027930620145297166845043047, −8.142979178357257791492590163475, −6.62004563997498246905211032079, −5.32301792161447471266208285493, −4.97763842017610172822721777672, −3.91832035106930854092957155338, −1.70353941363030166024034576213, 0.04327217044222701367822391631, 0.865159310004814737033317587456, 3.03657015182723284478150970470, 4.13099024624070704783644534916, 5.73621291877810297562346948493, 6.57025113510299097339271598517, 7.36946903083894407274877364383, 7.942565777627509131313437949182, 9.952931794847179876226802471039, 10.86374343803169146938676057055

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