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Properties

Label 2-304-19.18-c4-0-13
Degree $2$
Conductor $304$
Sign $0.0387 - 0.999i$
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  + 7.80i·3-s − 33.0·5-s − 16.0·7-s + 20.1·9-s + 215.·11-s − 281. i·13-s − 258. i·15-s + 226.·17-s + (−13.9 + 360. i)19-s − 125. i·21-s − 414.·23-s + 470.·25-s + 788. i·27-s + 606. i·29-s + 478. i·31-s + ⋯
L(s)  = 1  + 0.867i·3-s − 1.32·5-s − 0.328·7-s + 0.248·9-s + 1.78·11-s − 1.66i·13-s − 1.14i·15-s + 0.784·17-s + (−0.0387 + 0.999i)19-s − 0.284i·21-s − 0.783·23-s + 0.752·25-s + 1.08i·27-s + 0.721i·29-s + 0.498i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0387 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0387 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.0387 - 0.999i$
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: \((2,\ 304,\ (\ :2),\ 0.0387 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.481589514\)
\(L(\frac12)\) \(\approx\) \(1.481589514\)
\(L(3)\) 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 \( 1 + (13.9 - 360. i)T \)
good3 \( 1 - 7.80iT - 81T^{2} \)
5 \( 1 + 33.0T + 625T^{2} \)
7 \( 1 + 16.0T + 2.40e3T^{2} \)
11 \( 1 - 215.T + 1.46e4T^{2} \)
13 \( 1 + 281. iT - 2.85e4T^{2} \)
17 \( 1 - 226.T + 8.35e4T^{2} \)
23 \( 1 + 414.T + 2.79e5T^{2} \)
29 \( 1 - 606. iT - 7.07e5T^{2} \)
31 \( 1 - 478. iT - 9.23e5T^{2} \)
37 \( 1 + 104. iT - 1.87e6T^{2} \)
41 \( 1 - 1.89e3iT - 2.82e6T^{2} \)
43 \( 1 + 315.T + 3.41e6T^{2} \)
47 \( 1 - 474.T + 4.87e6T^{2} \)
53 \( 1 + 774. iT - 7.89e6T^{2} \)
59 \( 1 + 567. iT - 1.21e7T^{2} \)
61 \( 1 - 4.79e3T + 1.38e7T^{2} \)
67 \( 1 + 4.46e3iT - 2.01e7T^{2} \)
71 \( 1 - 7.63e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.39e3T + 2.83e7T^{2} \)
79 \( 1 - 9.70e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.05e4T + 4.74e7T^{2} \)
89 \( 1 + 1.02e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.54e4iT - 8.85e7T^{2} \)
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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

−11.29970924686665948950003688734, −10.28034153062417458328900516675, −9.609920224260867286012787349423, −8.407976573216613807532608718492, −7.62271446692606976719861868747, −6.41703794686583088141711943150, −5.08277761790400052245031040988, −3.76355963716108948883030974287, −3.53219766880119279440307117217, −1.04979942016977108948690162964, 0.59012520428167751114267346071, 1.87643370860342122263579182162, 3.74015218066557357044657888027, 4.36922522620036114779181853232, 6.35333151895815177461429700980, 6.94864659320720057616790081609, 7.77147102464079719570859923902, 8.891587001189958667222402656970, 9.729134326281541565627811080161, 11.32844102273981560687458519731

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