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Properties

Label 2-304-19.18-c4-0-38
Degree $2$
Conductor $304$
Sign $-0.801 - 0.598i$
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  − 15.6i·3-s + 6.30·5-s + 28.3·7-s − 163.·9-s + 76.2·11-s − 47.6i·13-s − 98.4i·15-s − 367.·17-s + (−289. − 215. i)19-s − 442. i·21-s − 415.·23-s − 585.·25-s + 1.28e3i·27-s − 1.22e3i·29-s + 329. i·31-s + ⋯
L(s)  = 1  − 1.73i·3-s + 0.252·5-s + 0.577·7-s − 2.01·9-s + 0.630·11-s − 0.281i·13-s − 0.437i·15-s − 1.27·17-s + (−0.801 − 0.598i)19-s − 1.00i·21-s − 0.785·23-s − 0.936·25-s + 1.75i·27-s − 1.45i·29-s + 0.342i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.801 - 0.598i$
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: \((2,\ 304,\ (\ :2),\ -0.801 - 0.598i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9878490728\)
\(L(\frac12)\) \(\approx\) \(0.9878490728\)
\(L(3)\) 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 \( 1 + (289. + 215. i)T \)
good3 \( 1 + 15.6iT - 81T^{2} \)
5 \( 1 - 6.30T + 625T^{2} \)
7 \( 1 - 28.3T + 2.40e3T^{2} \)
11 \( 1 - 76.2T + 1.46e4T^{2} \)
13 \( 1 + 47.6iT - 2.85e4T^{2} \)
17 \( 1 + 367.T + 8.35e4T^{2} \)
23 \( 1 + 415.T + 2.79e5T^{2} \)
29 \( 1 + 1.22e3iT - 7.07e5T^{2} \)
31 \( 1 - 329. iT - 9.23e5T^{2} \)
37 \( 1 + 295. iT - 1.87e6T^{2} \)
41 \( 1 - 1.31e3iT - 2.82e6T^{2} \)
43 \( 1 - 983.T + 3.41e6T^{2} \)
47 \( 1 - 1.07e3T + 4.87e6T^{2} \)
53 \( 1 - 4.33e3iT - 7.89e6T^{2} \)
59 \( 1 + 1.16e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.71e3T + 1.38e7T^{2} \)
67 \( 1 - 3.30e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.46e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.44e3T + 2.83e7T^{2} \)
79 \( 1 + 1.03e4iT - 3.89e7T^{2} \)
83 \( 1 - 3.96e3T + 4.74e7T^{2} \)
89 \( 1 + 8.68e3iT - 6.27e7T^{2} \)
97 \( 1 - 8.41e3iT - 8.85e7T^{2} \)
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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

−10.84215391374986386726862904471, −9.325875026842033843704611135238, −8.337355634343221521931166656435, −7.61159850303414294118283608658, −6.55211967303708043979969255189, −5.94036218041793212768326345124, −4.35890605821594468329266807099, −2.47336939902904491036940521562, −1.64350722882731450068243962333, −0.28179561404934660535600383036, 2.05608812819805900892875283362, 3.72124825470772883888092082218, 4.41208579483980048968183487838, 5.43149343645330674974291659344, 6.54801171846699680965209430227, 8.205924814481757758086242281507, 9.030047112233115368708068204677, 9.753068078490196436430802444236, 10.71876596163255869446208201475, 11.25926732180725104765566766552

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