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Properties

Label 1-304-304.117-r1-0-0
Degree $1$
Conductor $304$
Sign $0.976 - 0.215i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 + 0.766i)3-s + (0.342 − 0.939i)5-s + (0.5 − 0.866i)7-s + (−0.173 + 0.984i)9-s + (0.866 − 0.5i)11-s + (−0.642 + 0.766i)13-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)17-s + (0.984 − 0.173i)21-s + (0.939 − 0.342i)23-s + (−0.766 − 0.642i)25-s + (−0.866 + 0.5i)27-s + (0.984 + 0.173i)29-s + (0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.342 − 0.939i)5-s + (0.5 − 0.866i)7-s + (−0.173 + 0.984i)9-s + (0.866 − 0.5i)11-s + (−0.642 + 0.766i)13-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)17-s + (0.984 − 0.173i)21-s + (0.939 − 0.342i)23-s + (−0.766 − 0.642i)25-s + (−0.866 + 0.5i)27-s + (0.984 + 0.173i)29-s + (0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.976 - 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.932516207 - 0.3201942820i\)
\(L(\frac12)\) \(\approx\) \(2.932516207 - 0.3201942820i\)
\(L(1)\) \(\approx\) \(1.592152258 + 0.004225136561i\)
\(L(1)\) \(\approx\) \(1.592152258 + 0.004225136561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.642 + 0.766i)T \)
5 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.99308578475072764601691363338, −24.712588584726388367831434690790, −23.255185582981152862837119894970, −22.52225276581942453067059208450, −21.53925152149228465406383103114, −20.59583655288284391698975591567, −19.50658275176512529125388638285, −18.87816209943929153285389535071, −17.78239241029827493487456964632, −17.55900505371027928849816603661, −15.68495117088562202210347122881, −14.69141316105753593791187025170, −14.387725253513118970064416819328, −13.19570872907187492884870980011, −12.13388871572417993836909850162, −11.42014250794285734477471498230, −9.94810126744269590233908043213, −9.12882114523135237064040357645, −7.97281595233986246871281825356, −7.06279027951341598871064382264, −6.20373146085897672045914444829, −4.89416051558042596915325887434, −3.120189786166361518570986830731, −2.49499579890933112484512797903, −1.223451216342065032501894005037, 0.95693308811683288024852343938, 2.22371733816430552296147207098, 3.90339767880886587386007471751, 4.443174838582964145745300375935, 5.60401047755350144949296948871, 7.09963103340082995937753886003, 8.349285754487432375820016384994, 8.99742338638783263780716428045, 9.974504905747554228318661206710, 10.9138724319813506126027540797, 12.08435467054797869257137555023, 13.32711194872531872450253128740, 14.11990193389866910067909030185, 14.80352836593742247260556877320, 16.10653802828183468379080519652, 16.92080553445249298849033676757, 17.329144487555504683815324105699, 19.16958769871882292723061865187, 19.7397044261436310389667473084, 20.69374491930557872742573122713, 21.32070993605988996973807138516, 22.07651977067228549507679240706, 23.402487003613061961443627718594, 24.38275688141758195037135932191, 24.9592320605824605780160800293

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