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Properties

Label 1-367-367.47-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.993 + 0.116i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
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  + (0.229 − 0.973i)2-s + (0.128 + 0.991i)3-s + (−0.894 − 0.447i)4-s + (−0.716 + 0.697i)5-s + (0.994 + 0.102i)6-s + (0.328 + 0.944i)7-s + (−0.640 + 0.767i)8-s + (−0.967 + 0.254i)9-s + (0.514 + 0.857i)10-s + (−0.469 − 0.882i)11-s + (0.328 − 0.944i)12-s + (−0.967 + 0.254i)13-s + (0.994 − 0.102i)14-s + (−0.784 − 0.620i)15-s + (0.600 + 0.799i)16-s + (−0.0771 − 0.997i)17-s + ⋯
L(s)  = 1  + (0.229 − 0.973i)2-s + (0.128 + 0.991i)3-s + (−0.894 − 0.447i)4-s + (−0.716 + 0.697i)5-s + (0.994 + 0.102i)6-s + (0.328 + 0.944i)7-s + (−0.640 + 0.767i)8-s + (−0.967 + 0.254i)9-s + (0.514 + 0.857i)10-s + (−0.469 − 0.882i)11-s + (0.328 − 0.944i)12-s + (−0.967 + 0.254i)13-s + (0.994 − 0.102i)14-s + (−0.784 − 0.620i)15-s + (0.600 + 0.799i)16-s + (−0.0771 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.993 + 0.116i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.993 + 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004571241471 + 0.07801685526i\)
\(L(\frac12)\) \(\approx\) \(0.004571241471 + 0.07801685526i\)
\(L(1)\) \(\approx\) \(0.6812309494 + 0.01694960512i\)
\(L(1)\) \(\approx\) \(0.6812309494 + 0.01694960512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.229 - 0.973i)T \)
3 \( 1 + (0.128 + 0.991i)T \)
5 \( 1 + (-0.716 + 0.697i)T \)
7 \( 1 + (0.328 + 0.944i)T \)
11 \( 1 + (-0.469 - 0.882i)T \)
13 \( 1 + (-0.967 + 0.254i)T \)
17 \( 1 + (-0.0771 - 0.997i)T \)
19 \( 1 + (-0.998 - 0.0514i)T \)
23 \( 1 + (0.600 - 0.799i)T \)
29 \( 1 + (-0.279 - 0.960i)T \)
31 \( 1 + (-0.894 - 0.447i)T \)
37 \( 1 + (0.916 + 0.400i)T \)
41 \( 1 + (-0.469 + 0.882i)T \)
43 \( 1 + (-0.998 + 0.0514i)T \)
47 \( 1 + (0.815 + 0.579i)T \)
53 \( 1 + (-0.716 + 0.697i)T \)
59 \( 1 + (-0.935 + 0.352i)T \)
61 \( 1 + (0.994 - 0.102i)T \)
67 \( 1 + (-0.935 - 0.352i)T \)
71 \( 1 + (-0.894 + 0.447i)T \)
73 \( 1 + (-0.843 + 0.536i)T \)
79 \( 1 + (-0.784 + 0.620i)T \)
83 \( 1 + T \)
89 \( 1 + (0.815 + 0.579i)T \)
97 \( 1 + (0.600 - 0.799i)T \)
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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

−23.970951139172947596018845912770, −23.622642619759208140104817534014, −23.12423164964487794251704081814, −21.836604441470665986572552002665, −20.5424510951667863565448070804, −19.78340592813947068057614233869, −18.93658291874334508255647260792, −17.65955480337235117593714950625, −17.235752875177563249647593459097, −16.411662597230053920424721552212, −15.02008524218849297823543891728, −14.65946286529705085716009053732, −13.24976922314892393614766422634, −12.83846330136270179992373543496, −12.01705670030346295818790782009, −10.58824186444451924755431797422, −9.13346881154427746362982545138, −8.17221876858371848274564302647, −7.4609639091279576155627017084, −6.92418195166662295182327300541, −5.456473442551986167246333644054, −4.57107863651550220390491262301, −3.48440464190935131158639298412, −1.656031587317649235760070811215, −0.04181856250595414948107967491, 2.50990940183543259695794269243, 2.91368585262673315135455289614, 4.2314989107057734191477939035, 4.99402890698619492383839733760, 6.114045506360848144329664739752, 7.92915645210441755459349372544, 8.82362557352985610785091985420, 9.71289041281596982676971316931, 10.78171388806916715750327670026, 11.36850764296677070444318768012, 12.103384692241554042072763429722, 13.422058247349107544777027919495, 14.70579948652555410546831560842, 14.86944306914998994157433579758, 16.01103759162826835475098601756, 17.15370329660198665854562320467, 18.533696672493800273957504420292, 18.91752337246721372461511800076, 19.94188375355070832471429182857, 20.818010469215652270656681907199, 21.73973841421449593126734871107, 22.08523700694813352608254491136, 23.01866449665690477575441735993, 23.92910401607054113071820196886, 25.133216778534563617684578681176

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