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Properties

Label 1-367-367.52-r0-0-0
Degree $1$
Conductor $367$
Sign $0.196 + 0.980i$
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  + (−0.0771 + 0.997i)2-s + (−0.843 + 0.536i)3-s + (−0.988 − 0.153i)4-s + (−0.967 + 0.254i)5-s + (−0.469 − 0.882i)6-s + (0.916 + 0.400i)7-s + (0.229 − 0.973i)8-s + (0.423 − 0.905i)9-s + (−0.179 − 0.983i)10-s + (−0.935 − 0.352i)11-s + (0.916 − 0.400i)12-s + (0.423 − 0.905i)13-s + (−0.469 + 0.882i)14-s + (0.679 − 0.733i)15-s + (0.952 + 0.304i)16-s + (0.0257 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.0771 + 0.997i)2-s + (−0.843 + 0.536i)3-s + (−0.988 − 0.153i)4-s + (−0.967 + 0.254i)5-s + (−0.469 − 0.882i)6-s + (0.916 + 0.400i)7-s + (0.229 − 0.973i)8-s + (0.423 − 0.905i)9-s + (−0.179 − 0.983i)10-s + (−0.935 − 0.352i)11-s + (0.916 − 0.400i)12-s + (0.423 − 0.905i)13-s + (−0.469 + 0.882i)14-s + (0.679 − 0.733i)15-s + (0.952 + 0.304i)16-s + (0.0257 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.196 + 0.980i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5430988264 + 0.4451988868i\)
\(L(\frac12)\) \(\approx\) \(0.5430988264 + 0.4451988868i\)
\(L(1)\) \(\approx\) \(0.5676873941 + 0.3706608425i\)
\(L(1)\) \(\approx\) \(0.5676873941 + 0.3706608425i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.0771 + 0.997i)T \)
3 \( 1 + (-0.843 + 0.536i)T \)
5 \( 1 + (-0.967 + 0.254i)T \)
7 \( 1 + (0.916 + 0.400i)T \)
11 \( 1 + (-0.935 - 0.352i)T \)
13 \( 1 + (0.423 - 0.905i)T \)
17 \( 1 + (0.0257 + 0.999i)T \)
19 \( 1 + (0.514 - 0.857i)T \)
23 \( 1 + (0.952 - 0.304i)T \)
29 \( 1 + (0.815 - 0.579i)T \)
31 \( 1 + (-0.988 - 0.153i)T \)
37 \( 1 + (-0.376 - 0.926i)T \)
41 \( 1 + (-0.935 + 0.352i)T \)
43 \( 1 + (0.514 + 0.857i)T \)
47 \( 1 + (0.978 + 0.204i)T \)
53 \( 1 + (-0.967 + 0.254i)T \)
59 \( 1 + (0.600 + 0.799i)T \)
61 \( 1 + (-0.469 + 0.882i)T \)
67 \( 1 + (0.600 - 0.799i)T \)
71 \( 1 + (-0.988 + 0.153i)T \)
73 \( 1 + (0.328 - 0.944i)T \)
79 \( 1 + (0.679 + 0.733i)T \)
83 \( 1 + T \)
89 \( 1 + (0.978 + 0.204i)T \)
97 \( 1 + (0.952 - 0.304i)T \)
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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

−23.88560729652330515473463233410, −23.57474374801244674795070784197, −22.88119851304349138901053273517, −21.82984778003089555484890508774, −20.72801897588379151961981238056, −20.30478289641353874028094982812, −18.8909117142446057657766008714, −18.5948760158585069675930795678, −17.61565811198081566348423091920, −16.7044856418928204025403385613, −15.77131910644990347502039332288, −14.301142022196880088570539849881, −13.44574007497904919034228581642, −12.41735534483268716638586453195, −11.72750027322085017833979120820, −11.09400107686968086745624860758, −10.29042818520171248439871552039, −8.83314068187456937880279876599, −7.82412296767393785285320019766, −7.11161424045425413967095450025, −5.182906679339164852475831020652, −4.76622419626291786400895320792, −3.50792696141801756300919989771, −1.93401942040415234898805647013, −0.888882639272020728239651935666, 0.756672213328544554171997380741, 3.23908635040294757123526555881, 4.43540964535429039212720757961, 5.211396185765056730474291052426, 6.05827251245317220853544552718, 7.31990638263073497311743861677, 8.13487003722843051707644658301, 9.012088085940340841029990779540, 10.54580522764109372170389912045, 10.98863556947242050665599150745, 12.235781093167631889910121844026, 13.16833864459977407741212890148, 14.61582630792340457431709183540, 15.370306915240993134762527752209, 15.73955452827185193758034837791, 16.77014454449551151866080768979, 17.7930752591304278134787062336, 18.256367382312326069046091311710, 19.29665636151673879427459066418, 20.6982823469778924752223226264, 21.63380960658719629245527507774, 22.45193594044751475060702111034, 23.36237181185519961529760280737, 23.798374275407232927207119234692, 24.61109930256703962088659887446

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