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Properties

Label 1-365-365.158-r1-0-0
Degree $1$
Conductor $365$
Sign $-0.964 + 0.262i$
Analytic cond. $39.2246$
Root an. cond. $39.2246$
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  + (0.342 + 0.939i)2-s + (0.866 + 0.5i)3-s + (−0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.5 − 0.866i)7-s + (−0.866 − 0.5i)8-s + (0.5 + 0.866i)9-s + (0.342 − 0.939i)11-s + (−0.984 + 0.173i)12-s + (−0.173 + 0.984i)13-s + (0.984 + 0.173i)14-s + (0.173 − 0.984i)16-s + (0.5 + 0.866i)17-s + (−0.642 + 0.766i)18-s + (−0.939 + 0.342i)19-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (0.866 + 0.5i)3-s + (−0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.5 − 0.866i)7-s + (−0.866 − 0.5i)8-s + (0.5 + 0.866i)9-s + (0.342 − 0.939i)11-s + (−0.984 + 0.173i)12-s + (−0.173 + 0.984i)13-s + (0.984 + 0.173i)14-s + (0.173 − 0.984i)16-s + (0.5 + 0.866i)17-s + (−0.642 + 0.766i)18-s + (−0.939 + 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 365 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 365 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(365\)    =    \(5 \cdot 73\)
Sign: $-0.964 + 0.262i$
Analytic conductor: \(39.2246\)
Root analytic conductor: \(39.2246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{365} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 365,\ (1:\ ),\ -0.964 + 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3578866574 + 2.675478862i\)
\(L(\frac12)\) \(\approx\) \(0.3578866574 + 2.675478862i\)
\(L(1)\) \(\approx\) \(1.115054906 + 1.107904494i\)
\(L(1)\) \(\approx\) \(1.115054906 + 1.107904494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.342 - 0.939i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + (-0.642 + 0.766i)T \)
37 \( 1 + (-0.342 + 0.939i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−24.12575670229234333697531638544, −23.01495313001232933134212302896, −22.28472556106510099622899847107, −21.110493247591847265138319970596, −20.61452874737947512803837439202, −19.76205554203105380185216555767, −18.936818573524051953460892026222, −18.13099261178402192944764914613, −17.49198193509988554365752395799, −15.55198655122326255880824999108, −14.799258685295020877623703563727, −14.23060826736713692894913488675, −12.989417974897869864739457623517, −12.43812504276301001478843918588, −11.6268088902890163003389882913, −10.31192259213316708841485465850, −9.39360006114536843804789637986, −8.58210287509984762719487572024, −7.56936223527678046186597870465, −6.14688510591405909123421581391, −4.95277611986482888936467323679, −3.88386499457727357053802431191, −2.5548168406297048579505239413, −2.07973340448995341781589238241, −0.60068540284911067232532659995, 1.53241001738322511733965962446, 3.35215507741682345091574971628, 4.02436198100313786750585269985, 4.964067135883135185822272304052, 6.28854120998629311185230188026, 7.36455896302426116892982666975, 8.25180426793041045478919704726, 8.946821228204723179557407616620, 10.07339442780110075046830242244, 11.16305532057141758352982169865, 12.579749368572815917421576467343, 13.66835899463780114246608465445, 14.242416494351320725774917594454, 14.81330783380960902649711617045, 16.010751451863903572973565344233, 16.66625454577202244587555015327, 17.42365085777526016180373238730, 18.81485872218208606003707537638, 19.50363960213753031416146760952, 20.72827538256556197779215688391, 21.55809772767742836785606963005, 22.00205812779341973653517323706, 23.65771800239702085007789731357, 23.77820023625449337537214394431, 24.95117600723315227012459749340

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