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Properties

Label 1-367-367.145-r0-0-0
Degree $1$
Conductor $367$
Sign $0.384 + 0.923i$
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  + (−0.558 − 0.829i)2-s + (−0.988 − 0.153i)3-s + (−0.376 + 0.926i)4-s + (0.600 + 0.799i)5-s + (0.423 + 0.905i)6-s + (0.514 + 0.857i)7-s + (0.978 − 0.204i)8-s + (0.952 + 0.304i)9-s + (0.328 − 0.944i)10-s + (−0.784 + 0.620i)11-s + (0.514 − 0.857i)12-s + (0.952 + 0.304i)13-s + (0.423 − 0.905i)14-s + (−0.469 − 0.882i)15-s + (−0.716 − 0.697i)16-s + (0.751 + 0.660i)17-s + ⋯
L(s)  = 1  + (−0.558 − 0.829i)2-s + (−0.988 − 0.153i)3-s + (−0.376 + 0.926i)4-s + (0.600 + 0.799i)5-s + (0.423 + 0.905i)6-s + (0.514 + 0.857i)7-s + (0.978 − 0.204i)8-s + (0.952 + 0.304i)9-s + (0.328 − 0.944i)10-s + (−0.784 + 0.620i)11-s + (0.514 − 0.857i)12-s + (0.952 + 0.304i)13-s + (0.423 − 0.905i)14-s + (−0.469 − 0.882i)15-s + (−0.716 − 0.697i)16-s + (0.751 + 0.660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.384 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5539339206 + 0.3692419467i\)
\(L(\frac12)\) \(\approx\) \(0.5539339206 + 0.3692419467i\)
\(L(1)\) \(\approx\) \(0.6558903391 + 0.04063575383i\)
\(L(1)\) \(\approx\) \(0.6558903391 + 0.04063575383i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.558 - 0.829i)T \)
3 \( 1 + (-0.988 - 0.153i)T \)
5 \( 1 + (0.600 + 0.799i)T \)
7 \( 1 + (0.514 + 0.857i)T \)
11 \( 1 + (-0.784 + 0.620i)T \)
13 \( 1 + (0.952 + 0.304i)T \)
17 \( 1 + (0.751 + 0.660i)T \)
19 \( 1 + (-0.843 - 0.536i)T \)
23 \( 1 + (-0.716 + 0.697i)T \)
29 \( 1 + (0.0257 - 0.999i)T \)
31 \( 1 + (-0.376 + 0.926i)T \)
37 \( 1 + (-0.179 - 0.983i)T \)
41 \( 1 + (-0.784 - 0.620i)T \)
43 \( 1 + (-0.843 + 0.536i)T \)
47 \( 1 + (0.870 + 0.492i)T \)
53 \( 1 + (0.600 + 0.799i)T \)
59 \( 1 + (0.679 - 0.733i)T \)
61 \( 1 + (0.423 - 0.905i)T \)
67 \( 1 + (0.679 + 0.733i)T \)
71 \( 1 + (-0.376 - 0.926i)T \)
73 \( 1 + (-0.998 - 0.0514i)T \)
79 \( 1 + (-0.469 + 0.882i)T \)
83 \( 1 + T \)
89 \( 1 + (0.870 + 0.492i)T \)
97 \( 1 + (-0.716 + 0.697i)T \)
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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

−24.33430739405934549742774510840, −23.68347062320329574423541784478, −23.21558446403537704581937306715, −22.00610007227518766271013184575, −20.87639338213496414086594836174, −20.32516398694064757810119281083, −18.6451019153402735585093124005, −18.19196496398805055488742181849, −17.22241335637631761965149411395, −16.498104642083082211306376096003, −16.15947634533866071251706498341, −14.8529251889496067081296711911, −13.67099380711163112282219833236, −13.062046533947866050340857635845, −11.632854702974497274457223789047, −10.41181620862910772497013210595, −10.17527937467744663046308084620, −8.69774105484110504879806572720, −7.93301400703945454538980836772, −6.71099379271340159777824238366, −5.73206427037887574330755519012, −5.11957401600834698504165538699, −4.05580960987111758330885365845, −1.58707539407793457909447646445, −0.59041411000385522035715273121, 1.60272832495823450374743573421, 2.32591158566517477700322760956, 3.82741354899179640731576416334, 5.16775273366783820191824743860, 6.123880023899742668072921627227, 7.29325288659446182606590088127, 8.344927978504245546085763640985, 9.59618969079365589981171300430, 10.48555973161041571330881598232, 11.076927059240652721271126504309, 12.00147229338626894776683394396, 12.83526521489733372757636623074, 13.777334794034898934092374870652, 15.16490458248233219231854578319, 16.117924287448569537925356341125, 17.45695130351786032503737851908, 17.74777278620039901677086277244, 18.635790502965022299015197483516, 19.10122792287855899541011082729, 20.70744627137716600699046525376, 21.582661836016425631536110072752, 21.760419808833546393704967273082, 23.02564123344846629011525823112, 23.60719386568149943297092866331, 25.12837911573304025682464704451

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