Properties

Label 2-304-19.18-c4-0-19
Degree $2$
Conductor $304$
Sign $0.753 + 0.657i$
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  + 16.8i·3-s − 14.7·5-s − 59.6·7-s − 201.·9-s + 17.1·11-s + 163. i·13-s − 248. i·15-s + 177.·17-s + (−272. − 237. i)19-s − 1.00e3i·21-s + 502.·23-s − 406.·25-s − 2.03e3i·27-s − 500. i·29-s + 1.46e3i·31-s + ⋯
L(s)  = 1  + 1.86i·3-s − 0.590·5-s − 1.21·7-s − 2.49·9-s + 0.141·11-s + 0.966i·13-s − 1.10i·15-s + 0.614·17-s + (−0.753 − 0.657i)19-s − 2.27i·21-s + 0.949·23-s − 0.650·25-s − 2.79i·27-s − 0.595i·29-s + 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.753 + 0.657i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1671931694\)
\(L(\frac12)\) \(\approx\) \(0.1671931694\)
\(L(3)\) 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 \( 1 + (272. + 237. i)T \)
good3 \( 1 - 16.8iT - 81T^{2} \)
5 \( 1 + 14.7T + 625T^{2} \)
7 \( 1 + 59.6T + 2.40e3T^{2} \)
11 \( 1 - 17.1T + 1.46e4T^{2} \)
13 \( 1 - 163. iT - 2.85e4T^{2} \)
17 \( 1 - 177.T + 8.35e4T^{2} \)
23 \( 1 - 502.T + 2.79e5T^{2} \)
29 \( 1 + 500. iT - 7.07e5T^{2} \)
31 \( 1 - 1.46e3iT - 9.23e5T^{2} \)
37 \( 1 + 600. iT - 1.87e6T^{2} \)
41 \( 1 - 325. iT - 2.82e6T^{2} \)
43 \( 1 - 752.T + 3.41e6T^{2} \)
47 \( 1 - 2.25e3T + 4.87e6T^{2} \)
53 \( 1 - 1.50e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.55e3iT - 1.21e7T^{2} \)
61 \( 1 - 770.T + 1.38e7T^{2} \)
67 \( 1 + 1.61e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.67e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.33e3T + 2.83e7T^{2} \)
79 \( 1 + 53.3iT - 3.89e7T^{2} \)
83 \( 1 + 9.43e3T + 4.74e7T^{2} \)
89 \( 1 + 9.23e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.56e3iT - 8.85e7T^{2} \)
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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

−10.84930656458515153456912727344, −9.948067328550713222475884022701, −9.273134156983989422138816120094, −8.551392999790780191967935551364, −6.96567645820538918390800367044, −5.80887402305650930822460545202, −4.59114009077972828203548507805, −3.80516016835024293714567615105, −2.89807976016664205902446508767, −0.06281054938469694194333047115, 0.968792501342510269398009091666, 2.52324862913536761525375985235, 3.57345780723532813059484498856, 5.66855887360203556477159458117, 6.39968704896091100654125272360, 7.38153999516357350519600133896, 7.989726326087647930885169161324, 9.039389945464227390373916139601, 10.36113560459101739888171263950, 11.54106216101716849579287397244

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