Properties

Label 1-368-368.291-r0-0-0
Degree $1$
Conductor $368$
Sign $0.0493 + 0.998i$
Analytic cond. $1.70898$
Root an. cond. $1.70898$
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  + (0.755 + 0.654i)3-s + (−0.989 − 0.142i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (−0.654 − 0.755i)15-s + (−0.841 + 0.540i)17-s + (−0.540 + 0.841i)19-s + (0.281 − 0.959i)21-s + (0.959 + 0.281i)25-s + (−0.540 + 0.841i)27-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (−0.415 + 0.909i)33-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (−0.989 − 0.142i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (−0.654 − 0.755i)15-s + (−0.841 + 0.540i)17-s + (−0.540 + 0.841i)19-s + (0.281 − 0.959i)21-s + (0.959 + 0.281i)25-s + (−0.540 + 0.841i)27-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (−0.415 + 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0493 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0493 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.0493 + 0.998i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.0493 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8886318944 + 0.8457784477i\)
\(L(\frac12)\) \(\approx\) \(0.8886318944 + 0.8457784477i\)
\(L(1)\) \(\approx\) \(1.030791364 + 0.3608360419i\)
\(L(1)\) \(\approx\) \(1.030791364 + 0.3608360419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.281 + 0.959i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 + (0.540 + 0.841i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (0.989 - 0.142i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.909 + 0.415i)T \)
61 \( 1 + (-0.755 + 0.654i)T \)
67 \( 1 + (0.281 - 0.959i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−24.51377155690740535295370448448, −23.71053847072619935450427121518, −22.80932756386284887873943208199, −21.85111041312846081054978629638, −20.78179522043605597617285870404, −19.82754807493562612782359302371, −19.15036452412405401207015027429, −18.58480436672082999721537670281, −17.63524795964632526095757165206, −16.08720882445141197823321607981, −15.524842272331365563865490792444, −14.71474193508955976830326955239, −13.50897783473294173631225971327, −12.89535283232728874309232692420, −11.72997612018030487276855010162, −11.16287165227774651639850813505, −9.5003244752220816701402703716, −8.561034386930667074590131335383, −8.0960972974780944526114498808, −6.75088068695133444187195254613, −6.06120019933530511126694370317, −4.35114156625583542579562438880, −3.22782212865996009685789261840, −2.495850646580933858667727974755, −0.716496283518734661621382122225, 1.57314780465736078058284886498, 3.16948375681456518957578565839, 4.12191421931111153479128963212, 4.54143938210842255868344943513, 6.44446337403612188672311959298, 7.44730050744933018330995932233, 8.36553088289996574066966222516, 9.223702619474150405157161254885, 10.374976945590394847841268479500, 11.00483924594781286231656122020, 12.348386796241636987805529453656, 13.251641437347774204111813043976, 14.29316790705806637972310102606, 15.07718675148549208828274603524, 16.00935987173473491037371141819, 16.53544693905980909397267391480, 17.75586951953949529519825703108, 19.13410151538811414456930408176, 19.68276704774515846638634357027, 20.40315922994498208777039673600, 21.11642179796628392652121769332, 22.36575043492584824649252655974, 23.11772367683559255492595402446, 23.84243873578566137576972319633, 25.06311033760991219408453913812

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