Properties

Label 2-304-19.8-c2-0-3
Degree $2$
Conductor $304$
Sign $0.0186 - 0.999i$
Analytic cond. $8.28340$
Root an. cond. $2.87808$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.72 − 1.57i)3-s + (0.5 − 0.866i)5-s − 6.89·7-s + (0.449 + 0.778i)9-s + 14.8·11-s + (−14.8 + 8.57i)13-s + (−2.72 + 1.57i)15-s + (1.05 − 1.81i)17-s + (−11.3 + 15.2i)19-s + (18.7 + 10.8i)21-s + (13.5 + 23.4i)23-s + (12 + 20.7i)25-s + 25.4i·27-s + (−5.54 + 3.20i)29-s − 31.1i·31-s + ⋯
L(s)  = 1  + (−0.908 − 0.524i)3-s + (0.100 − 0.173i)5-s − 0.985·7-s + (0.0499 + 0.0865i)9-s + 1.35·11-s + (−1.14 + 0.659i)13-s + (−0.181 + 0.104i)15-s + (0.0617 − 0.107i)17-s + (−0.597 + 0.802i)19-s + (0.895 + 0.516i)21-s + (0.587 + 1.01i)23-s + (0.479 + 0.831i)25-s + 0.943i·27-s + (−0.191 + 0.110i)29-s − 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0186 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0186 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.0186 - 0.999i$
Analytic conductor: \(8.28340\)
Root analytic conductor: \(2.87808\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1),\ 0.0186 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.380523 + 0.373501i\)
\(L(\frac12)\) \(\approx\) \(0.380523 + 0.373501i\)
\(L(2)\) 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 \( 1 + (11.3 - 15.2i)T \)
good3 \( 1 + (2.72 + 1.57i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 6.89T + 49T^{2} \)
11 \( 1 - 14.8T + 121T^{2} \)
13 \( 1 + (14.8 - 8.57i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-1.05 + 1.81i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-13.5 - 23.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (5.54 - 3.20i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 - 28.9iT - 1.36e3T^{2} \)
41 \( 1 + (-55.9 - 32.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (37.6 - 65.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (5.77 + 9.99i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (69.2 - 40.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (50.9 + 29.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.09 + 1.89i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-51.6 + 29.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (87.5 + 50.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-63.6 + 110. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-5.78 - 3.33i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 1.30T + 6.88e3T^{2} \)
89 \( 1 + (-5.84 + 3.37i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (114. + 65.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.79960735477417133901361464022, −11.09407652309369447192545409523, −9.555924095115784345224240182779, −9.328974479592194295143278651348, −7.61954888245875700108448328265, −6.58616266788696888014773460101, −6.11157304712838105644859695715, −4.74985215152928088381888849199, −3.34173305790262830514802792830, −1.41569306446485322127542838231, 0.30096569446447775367454939932, 2.69455622611018578918757335415, 4.16999249029470649433325052208, 5.22215289309745251418540249291, 6.35848588296310429413494098485, 7.01315247781292169415759799247, 8.624950673111656394074336046239, 9.597697511553098427275066908302, 10.40561617648824566508207367062, 11.12277233232843247291542792516

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