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Properties

Label 1-367-367.25-r0-0-0
Degree $1$
Conductor $367$
Sign $0.908 - 0.417i$
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  + (0.978 + 0.204i)2-s + (−0.894 + 0.447i)3-s + (0.916 + 0.400i)4-s + (−0.935 − 0.352i)5-s + (−0.967 + 0.254i)6-s + (−0.998 − 0.0514i)7-s + (0.815 + 0.579i)8-s + (0.600 − 0.799i)9-s + (−0.843 − 0.536i)10-s + (0.423 − 0.905i)11-s + (−0.998 + 0.0514i)12-s + (0.600 − 0.799i)13-s + (−0.967 − 0.254i)14-s + (0.994 − 0.102i)15-s + (0.679 + 0.733i)16-s + (−0.558 − 0.829i)17-s + ⋯
L(s)  = 1  + (0.978 + 0.204i)2-s + (−0.894 + 0.447i)3-s + (0.916 + 0.400i)4-s + (−0.935 − 0.352i)5-s + (−0.967 + 0.254i)6-s + (−0.998 − 0.0514i)7-s + (0.815 + 0.579i)8-s + (0.600 − 0.799i)9-s + (−0.843 − 0.536i)10-s + (0.423 − 0.905i)11-s + (−0.998 + 0.0514i)12-s + (0.600 − 0.799i)13-s + (−0.967 − 0.254i)14-s + (0.994 − 0.102i)15-s + (0.679 + 0.733i)16-s + (−0.558 − 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.326308818 - 0.2901989646i\)
\(L(\frac12)\) \(\approx\) \(1.326308818 - 0.2901989646i\)
\(L(1)\) \(\approx\) \(1.201944322 + 0.03245875389i\)
\(L(1)\) \(\approx\) \(1.201944322 + 0.03245875389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.978 + 0.204i)T \)
3 \( 1 + (-0.894 + 0.447i)T \)
5 \( 1 + (-0.935 - 0.352i)T \)
7 \( 1 + (-0.998 - 0.0514i)T \)
11 \( 1 + (0.423 - 0.905i)T \)
13 \( 1 + (0.600 - 0.799i)T \)
17 \( 1 + (-0.558 - 0.829i)T \)
19 \( 1 + (0.128 + 0.991i)T \)
23 \( 1 + (0.679 - 0.733i)T \)
29 \( 1 + (-0.0771 - 0.997i)T \)
31 \( 1 + (0.916 + 0.400i)T \)
37 \( 1 + (0.514 - 0.857i)T \)
41 \( 1 + (0.423 + 0.905i)T \)
43 \( 1 + (0.128 - 0.991i)T \)
47 \( 1 + (0.0257 - 0.999i)T \)
53 \( 1 + (-0.935 - 0.352i)T \)
59 \( 1 + (-0.784 + 0.620i)T \)
61 \( 1 + (-0.967 - 0.254i)T \)
67 \( 1 + (-0.784 - 0.620i)T \)
71 \( 1 + (0.916 - 0.400i)T \)
73 \( 1 + (-0.988 + 0.153i)T \)
79 \( 1 + (0.994 + 0.102i)T \)
83 \( 1 + T \)
89 \( 1 + (0.0257 - 0.999i)T \)
97 \( 1 + (0.679 - 0.733i)T \)
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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

−24.27708692359826915846679767838, −23.60525088018522423942069527727, −22.99922838903302048766106029251, −22.30595259520470514035909225512, −21.66476718098863871021280857433, −20.25941769147532730824621567891, −19.37057208140980348047320655424, −18.95429504081465710284257588575, −17.57702736445549564907133369098, −16.49223445549101150725440053428, −15.717118958415909105792628828065, −15.06007603539900340546048794120, −13.70171297639522345281967735160, −12.852694716881979056491612133396, −12.170950551957012941202048422870, −11.31290111717580028071756967044, −10.6750575689531636486425219086, −9.36003869391041350716245041041, −7.58251865842895550721315997142, −6.7049223968360893489795371775, −6.28551766289816101937475529618, −4.77716150877254837740003026371, −4.03193039367912395688515872170, −2.81158465097927990322656862090, −1.35798984289076786215615537165, 0.74944682089889733732898732702, 3.092646912218569046166569998791, 3.81207659571338264684525520566, 4.76503131746672108159655168639, 5.896520315987943452229546417102, 6.5397720627633102073627155827, 7.720698792574333417445888441259, 8.96950415776796029674234603389, 10.40117217821006708389275915154, 11.23035501873965193270040582604, 12.03099593484215835752987773911, 12.76920074823158433637661516949, 13.69370458499638092315445146226, 15.07278026974310292237820265475, 15.80312257778594524197229703058, 16.35186212317753635387038649573, 16.995047017305547532125342549338, 18.47983108950780753302864638042, 19.53765635115592075082913956552, 20.46191193848038400248890812040, 21.24478481274781874515942157702, 22.39536685843770888957614910918, 22.854982094567243954891153633, 23.33781667839756163557583541360, 24.51609046560885142371298389876

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