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Properties

Label 2-304-16.13-c1-0-28
Degree $2$
Conductor $304$
Sign $0.993 - 0.113i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.701i)2-s + (2.26 + 2.26i)3-s + (1.01 − 1.72i)4-s + (−1.08 + 1.08i)5-s + (4.37 + 1.19i)6-s − 4.17i·7-s + (0.0356 − 2.82i)8-s + 7.29i·9-s + (−0.572 + 2.10i)10-s + (−0.984 + 0.984i)11-s + (6.21 − 1.60i)12-s + (−0.198 − 0.198i)13-s + (−2.93 − 5.12i)14-s − 4.94·15-s + (−1.94 − 3.49i)16-s − 4.10·17-s + ⋯
L(s)  = 1  + (0.868 − 0.496i)2-s + (1.31 + 1.31i)3-s + (0.507 − 0.861i)4-s + (−0.487 + 0.487i)5-s + (1.78 + 0.487i)6-s − 1.57i·7-s + (0.0126 − 0.999i)8-s + 2.43i·9-s + (−0.181 + 0.665i)10-s + (−0.296 + 0.296i)11-s + (1.79 − 0.464i)12-s + (−0.0549 − 0.0549i)13-s + (−0.783 − 1.37i)14-s − 1.27·15-s + (−0.485 − 0.874i)16-s − 0.996·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.993 - 0.113i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.993 - 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71474 + 0.155042i\)
\(L(\frac12)\) \(\approx\) \(2.71474 + 0.155042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.701i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-2.26 - 2.26i)T + 3iT^{2} \)
5 \( 1 + (1.08 - 1.08i)T - 5iT^{2} \)
7 \( 1 + 4.17iT - 7T^{2} \)
11 \( 1 + (0.984 - 0.984i)T - 11iT^{2} \)
13 \( 1 + (0.198 + 0.198i)T + 13iT^{2} \)
17 \( 1 + 4.10T + 17T^{2} \)
23 \( 1 - 7.37iT - 23T^{2} \)
29 \( 1 + (4.06 + 4.06i)T + 29iT^{2} \)
31 \( 1 - 0.696T + 31T^{2} \)
37 \( 1 + (-6.01 + 6.01i)T - 37iT^{2} \)
41 \( 1 + 3.09iT - 41T^{2} \)
43 \( 1 + (-6.73 + 6.73i)T - 43iT^{2} \)
47 \( 1 + 6.28T + 47T^{2} \)
53 \( 1 + (-3.69 + 3.69i)T - 53iT^{2} \)
59 \( 1 + (-5.17 + 5.17i)T - 59iT^{2} \)
61 \( 1 + (-3.00 - 3.00i)T + 61iT^{2} \)
67 \( 1 + (-7.34 - 7.34i)T + 67iT^{2} \)
71 \( 1 - 7.00iT - 71T^{2} \)
73 \( 1 + 7.46iT - 73T^{2} \)
79 \( 1 - 2.46T + 79T^{2} \)
83 \( 1 + (-7.91 - 7.91i)T + 83iT^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−11.29008526890994893978146330298, −10.85778502116772904369559196022, −10.00309990954903259384317477719, −9.311724958124028862927882315857, −7.78561759317924054079073653819, −7.08196045522925669698537920211, −5.18045217015266845734014933109, −3.91036619156667283427311476244, −3.82834652364890712854366119837, −2.37082868386827551571688490524, 2.20980703589333869322968288787, 2.97929513074552316098804464943, 4.50829415960961274759729708463, 6.00410177474147287561023766264, 6.77860817684235042265451707164, 8.058337955427048497726689023893, 8.425361761048899963010054010811, 9.174419270472215799576108332025, 11.35163256562128692344902702439, 12.26402293710710243098812229048

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