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Properties

Label 1-367-367.170-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.890 + 0.455i$
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  + (−0.957 − 0.287i)2-s + (0.128 − 0.991i)3-s + (0.834 + 0.551i)4-s + (−0.716 − 0.697i)5-s + (−0.408 + 0.912i)6-s + (0.328 − 0.944i)7-s + (−0.640 − 0.767i)8-s + (−0.967 − 0.254i)9-s + (0.485 + 0.874i)10-s + (−0.529 − 0.848i)11-s + (0.653 − 0.756i)12-s + (0.704 − 0.710i)13-s + (−0.586 + 0.810i)14-s + (−0.784 + 0.620i)15-s + (0.392 + 0.919i)16-s + (−0.824 − 0.565i)17-s + ⋯
L(s)  = 1  + (−0.957 − 0.287i)2-s + (0.128 − 0.991i)3-s + (0.834 + 0.551i)4-s + (−0.716 − 0.697i)5-s + (−0.408 + 0.912i)6-s + (0.328 − 0.944i)7-s + (−0.640 − 0.767i)8-s + (−0.967 − 0.254i)9-s + (0.485 + 0.874i)10-s + (−0.529 − 0.848i)11-s + (0.653 − 0.756i)12-s + (0.704 − 0.710i)13-s + (−0.586 + 0.810i)14-s + (−0.784 + 0.620i)15-s + (0.392 + 0.919i)16-s + (−0.824 − 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.890 + 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1335337129 - 0.5541674343i\)
\(L(\frac12)\) \(\approx\) \(-0.1335337129 - 0.5541674343i\)
\(L(1)\) \(\approx\) \(0.4004571158 - 0.4595606457i\)
\(L(1)\) \(\approx\) \(0.4004571158 - 0.4595606457i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.957 - 0.287i)T \)
3 \( 1 + (0.128 - 0.991i)T \)
5 \( 1 + (-0.716 - 0.697i)T \)
7 \( 1 + (0.328 - 0.944i)T \)
11 \( 1 + (-0.529 - 0.848i)T \)
13 \( 1 + (0.704 - 0.710i)T \)
17 \( 1 + (-0.824 - 0.565i)T \)
19 \( 1 + (0.543 + 0.839i)T \)
23 \( 1 + (0.392 - 0.919i)T \)
29 \( 1 + (-0.279 + 0.960i)T \)
31 \( 1 + (0.0600 - 0.998i)T \)
37 \( 1 + (-0.804 - 0.593i)T \)
41 \( 1 + (0.999 + 0.0343i)T \)
43 \( 1 + (0.454 + 0.890i)T \)
47 \( 1 + (0.815 - 0.579i)T \)
53 \( 1 + (-0.246 + 0.969i)T \)
59 \( 1 + (-0.935 - 0.352i)T \)
61 \( 1 + (-0.408 - 0.912i)T \)
67 \( 1 + (0.773 + 0.633i)T \)
71 \( 1 + (0.0600 + 0.998i)T \)
73 \( 1 + (0.886 - 0.462i)T \)
79 \( 1 + (-0.145 + 0.989i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.909 - 0.416i)T \)
97 \( 1 + (-0.992 + 0.119i)T \)
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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

−25.74993319595234845868988875735, −24.40499508661134842987507143616, −23.4903368159136458055038846673, −22.53572805682511447726445151269, −21.50120027594760162330741382098, −20.74292558079023664089337014478, −19.72674040498704678091750458096, −19.03491436117224976677513527910, −18.01955191288131153283488384314, −17.35471105789038780638406208568, −15.93636124555993011537540099700, −15.50905623187709487873170298727, −15.04355464868530151903821768914, −13.91646154914560734707444490820, −12.054845623084924988923602974646, −11.23276151984694702816770906788, −10.655854231353988129560432958638, −9.473078587423706559512516370931, −8.78733864063809420596448270179, −7.84583785050414861899691857447, −6.76737941785577581711638324623, −5.605329997380296579953969869700, −4.460133514131983247222463046070, −3.05907124380978848760697479190, −2.05274278809984782412742808043, 0.49782985944931171043543176269, 1.30869370065203920350849320260, 2.8238605986369838221542889839, 3.89735947623125854702232722178, 5.59571607425213705682165294935, 6.88072292336708073602909357277, 7.803531724330367281776109668736, 8.265536680483670209070065807, 9.20652962975918258028387219193, 10.83163699232090198974793709122, 11.20121104679298833375804312279, 12.42019652065133589854583547475, 13.082946844395816131224183206267, 14.11082112286641826428250481374, 15.57624552150577727929260515632, 16.42733164977330843952493792289, 17.16082439504727701984456383623, 18.20252239179689007416195640599, 18.75585272999828375010876960373, 19.82325671873031981180852339613, 20.3722221513418040962866149484, 20.933222777995956643520433998342, 22.657929618600046640286647346556, 23.520473751478484202426787026831, 24.49440895767188466687039063396

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