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Properties

Label 2-304-19.18-c6-0-31
Degree $2$
Conductor $304$
Sign $-0.333 + 0.942i$
Analytic cond. $69.9364$
Root an. cond. $8.36280$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 48.7i·3-s − 200.·5-s + 536.·7-s − 1.64e3·9-s + 528.·11-s + 1.93e3i·13-s + 9.75e3i·15-s + 6.54e3·17-s + (2.28e3 − 6.46e3i)19-s − 2.61e4i·21-s + 1.88e4·23-s + 2.43e4·25-s + 4.48e4i·27-s + 2.13e4i·29-s − 4.17e4i·31-s + ⋯
L(s)  = 1  − 1.80i·3-s − 1.60·5-s + 1.56·7-s − 2.26·9-s + 0.397·11-s + 0.882i·13-s + 2.88i·15-s + 1.33·17-s + (0.333 − 0.942i)19-s − 2.82i·21-s + 1.55·23-s + 1.56·25-s + 2.27i·27-s + 0.877i·29-s − 1.40i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.333 + 0.942i$
Analytic conductor: \(69.9364\)
Root analytic conductor: \(8.36280\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :3),\ -0.333 + 0.942i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.993130441\)
\(L(\frac12)\) \(\approx\) \(1.993130441\)
\(L(4)\) 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 \( 1 + (-2.28e3 + 6.46e3i)T \)
good3 \( 1 + 48.7iT - 729T^{2} \)
5 \( 1 + 200.T + 1.56e4T^{2} \)
7 \( 1 - 536.T + 1.17e5T^{2} \)
11 \( 1 - 528.T + 1.77e6T^{2} \)
13 \( 1 - 1.93e3iT - 4.82e6T^{2} \)
17 \( 1 - 6.54e3T + 2.41e7T^{2} \)
23 \( 1 - 1.88e4T + 1.48e8T^{2} \)
29 \( 1 - 2.13e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.17e4iT - 8.87e8T^{2} \)
37 \( 1 - 2.51e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.36e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.11e4T + 6.32e9T^{2} \)
47 \( 1 - 1.72e5T + 1.07e10T^{2} \)
53 \( 1 - 1.20e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.09e4iT - 4.21e10T^{2} \)
61 \( 1 - 9.64e4T + 5.15e10T^{2} \)
67 \( 1 - 3.93e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.59e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.44e4T + 1.51e11T^{2} \)
79 \( 1 - 4.29e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.05e6T + 3.26e11T^{2} \)
89 \( 1 + 4.42e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.50e5iT - 8.32e11T^{2} \)
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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

−11.11102334202334913592491538840, −8.897850363875114120825132158180, −8.223315236381885206756976662309, −7.34106809586429074209127564109, −7.10249743513851578927477502618, −5.48045364633437998818639415130, −4.28738523523505770988370316912, −2.81836642571512979370286565162, −1.40018730452192922609265040218, −0.73416042885361923835947072553, 0.881374846482822955032534507337, 3.17510574086058840439787800846, 3.89059764570121139510249259031, 4.81236907068700980230721324929, 5.47100599250356026201870937444, 7.57230185495747218464334280909, 8.190471621900315245734960504678, 9.013547707497440885219509776786, 10.29787957195058485158814547297, 10.90531522354473769959334825713

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