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Properties

Label 1-367-367.72-r0-0-0
Degree $1$
Conductor $367$
Sign $0.965 + 0.261i$
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  + (−0.935 − 0.352i)2-s + (0.0257 + 0.999i)3-s + (0.751 + 0.660i)4-s + (−0.988 + 0.153i)5-s + (0.328 − 0.944i)6-s + (−0.640 − 0.767i)7-s + (−0.469 − 0.882i)8-s + (−0.998 + 0.0514i)9-s + (0.978 + 0.204i)10-s + (0.916 − 0.400i)11-s + (−0.640 + 0.767i)12-s + (−0.998 + 0.0514i)13-s + (0.328 + 0.944i)14-s + (−0.179 − 0.983i)15-s + (0.128 + 0.991i)16-s + (0.600 + 0.799i)17-s + ⋯
L(s)  = 1  + (−0.935 − 0.352i)2-s + (0.0257 + 0.999i)3-s + (0.751 + 0.660i)4-s + (−0.988 + 0.153i)5-s + (0.328 − 0.944i)6-s + (−0.640 − 0.767i)7-s + (−0.469 − 0.882i)8-s + (−0.998 + 0.0514i)9-s + (0.978 + 0.204i)10-s + (0.916 − 0.400i)11-s + (−0.640 + 0.767i)12-s + (−0.998 + 0.0514i)13-s + (0.328 + 0.944i)14-s + (−0.179 − 0.983i)15-s + (0.128 + 0.991i)16-s + (0.600 + 0.799i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.965 + 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5823251619 + 0.07756243126i\)
\(L(\frac12)\) \(\approx\) \(0.5823251619 + 0.07756243126i\)
\(L(1)\) \(\approx\) \(0.5780778623 + 0.06150474885i\)
\(L(1)\) \(\approx\) \(0.5780778623 + 0.06150474885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.935 - 0.352i)T \)
3 \( 1 + (0.0257 + 0.999i)T \)
5 \( 1 + (-0.988 + 0.153i)T \)
7 \( 1 + (-0.640 - 0.767i)T \)
11 \( 1 + (0.916 - 0.400i)T \)
13 \( 1 + (-0.998 + 0.0514i)T \)
17 \( 1 + (0.600 + 0.799i)T \)
19 \( 1 + (0.815 - 0.579i)T \)
23 \( 1 + (0.128 - 0.991i)T \)
29 \( 1 + (-0.967 - 0.254i)T \)
31 \( 1 + (0.751 + 0.660i)T \)
37 \( 1 + (0.229 + 0.973i)T \)
41 \( 1 + (0.916 + 0.400i)T \)
43 \( 1 + (0.815 + 0.579i)T \)
47 \( 1 + (0.423 - 0.905i)T \)
53 \( 1 + (-0.988 + 0.153i)T \)
59 \( 1 + (-0.376 - 0.926i)T \)
61 \( 1 + (0.328 + 0.944i)T \)
67 \( 1 + (-0.376 + 0.926i)T \)
71 \( 1 + (0.751 - 0.660i)T \)
73 \( 1 + (0.870 + 0.492i)T \)
79 \( 1 + (-0.179 + 0.983i)T \)
83 \( 1 + T \)
89 \( 1 + (0.423 - 0.905i)T \)
97 \( 1 + (0.128 - 0.991i)T \)
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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

−24.72299203167441176914103542889, −24.12844448772793357892296626541, −22.98482801676067176337299297816, −22.43013429921686320614724140166, −20.64066700842712119978650970655, −19.79163135309387070550442062411, −19.21127764724840382964484717311, −18.64856473084530520977134171558, −17.619477628829973449600473057010, −16.75843712585971586283268795124, −15.883236176088561703613195373373, −14.93285586165998374723786530974, −14.12378082892506947268545484024, −12.46579781277225587018726941309, −12.02026425059396808147220766731, −11.233708048443469133748393020688, −9.543144974801445896477233683251, −9.08924874524880112539770107294, −7.59479956412055613929660154036, −7.492295058691774029981181418280, −6.24718759494363459788733989614, −5.26356742016575305389810310254, −3.341562753377549065088387040104, −2.190434038900849967375007758975, −0.807526749790573200629034714482, 0.755208563808640460415694823769, 2.85849269325969198919134731620, 3.61685467469220487884756981024, 4.50487199284361244350001837411, 6.3032596528356666664462802172, 7.32459070282831030193862244437, 8.28324836444558998883088463580, 9.312616912736477516094018832845, 10.06111031326403614851877586465, 10.93504974698451836178925509587, 11.70402267914388890746660508655, 12.62337528457546674634335900482, 14.23485068319649101810134907400, 15.13587376396563953006049852490, 16.08498817905553306857661943917, 16.736440762434630693919325687357, 17.3141916516995964218775143862, 18.84546468682625838825862070310, 19.64672232865181211358907953931, 19.9877635455000818097625085730, 20.98815027575759884342113318401, 22.141713253442007999667384166759, 22.58266962254339047505232048192, 23.88628484241007105303327305435, 24.89553441768206369300956161437

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