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Properties

Label 1-367-367.87-r0-0-0
Degree $1$
Conductor $367$
Sign $0.105 + 0.994i$
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  + (−0.988 + 0.153i)2-s + (0.423 + 0.905i)3-s + (0.952 − 0.304i)4-s + (0.870 + 0.492i)5-s + (−0.558 − 0.829i)6-s + (0.679 − 0.733i)7-s + (−0.894 + 0.447i)8-s + (−0.640 + 0.767i)9-s + (−0.935 − 0.352i)10-s + (0.751 − 0.660i)11-s + (0.679 + 0.733i)12-s + (−0.640 + 0.767i)13-s + (−0.558 + 0.829i)14-s + (−0.0771 + 0.997i)15-s + (0.815 − 0.579i)16-s + (−0.998 − 0.0514i)17-s + ⋯
L(s)  = 1  + (−0.988 + 0.153i)2-s + (0.423 + 0.905i)3-s + (0.952 − 0.304i)4-s + (0.870 + 0.492i)5-s + (−0.558 − 0.829i)6-s + (0.679 − 0.733i)7-s + (−0.894 + 0.447i)8-s + (−0.640 + 0.767i)9-s + (−0.935 − 0.352i)10-s + (0.751 − 0.660i)11-s + (0.679 + 0.733i)12-s + (−0.640 + 0.767i)13-s + (−0.558 + 0.829i)14-s + (−0.0771 + 0.997i)15-s + (0.815 − 0.579i)16-s + (−0.998 − 0.0514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.105 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8716361485 + 0.7837791557i\)
\(L(\frac12)\) \(\approx\) \(0.8716361485 + 0.7837791557i\)
\(L(1)\) \(\approx\) \(0.8844854118 + 0.4113383007i\)
\(L(1)\) \(\approx\) \(0.8844854118 + 0.4113383007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.988 + 0.153i)T \)
3 \( 1 + (0.423 + 0.905i)T \)
5 \( 1 + (0.870 + 0.492i)T \)
7 \( 1 + (0.679 - 0.733i)T \)
11 \( 1 + (0.751 - 0.660i)T \)
13 \( 1 + (-0.640 + 0.767i)T \)
17 \( 1 + (-0.998 - 0.0514i)T \)
19 \( 1 + (-0.469 + 0.882i)T \)
23 \( 1 + (0.815 + 0.579i)T \)
29 \( 1 + (0.328 + 0.944i)T \)
31 \( 1 + (0.952 - 0.304i)T \)
37 \( 1 + (-0.716 - 0.697i)T \)
41 \( 1 + (0.751 + 0.660i)T \)
43 \( 1 + (-0.469 - 0.882i)T \)
47 \( 1 + (0.916 - 0.400i)T \)
53 \( 1 + (0.870 + 0.492i)T \)
59 \( 1 + (-0.279 - 0.960i)T \)
61 \( 1 + (-0.558 + 0.829i)T \)
67 \( 1 + (-0.279 + 0.960i)T \)
71 \( 1 + (0.952 + 0.304i)T \)
73 \( 1 + (-0.784 + 0.620i)T \)
79 \( 1 + (-0.0771 - 0.997i)T \)
83 \( 1 + T \)
89 \( 1 + (0.916 - 0.400i)T \)
97 \( 1 + (0.815 + 0.579i)T \)
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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

−24.53581731191964096128984636520, −24.37527916588718284245201437846, −22.651167587247843466874766493463, −21.51781593977633741617583805133, −20.70664076487991954845584362554, −19.92140141412673853298926753440, −19.19363418999492329483829554265, −18.08828655998040184916826639384, −17.49217588457891036430568617867, −17.14024663282944076956662152335, −15.41798989044033786478063861040, −14.80857111931679360654423413799, −13.536455481441124056810546999698, −12.51823950733288575966992789861, −11.94528971438949335783854533512, −10.73929107431478451921967765567, −9.461064613345531907388532984225, −8.85823671748895490132133768251, −8.09133531764933562443993445160, −6.902654418701774822537821467608, −6.150725249410458205219502264250, −4.76441366603798692641464585795, −2.65859579931881735712034781876, −2.1282168796612302240077994331, −1.000088094981471220145344014819, 1.54081066697264151745080796516, 2.58639972743269346666677648419, 3.91451725912013643513554682829, 5.23305595227533296515867159184, 6.44551235452473388988864834710, 7.37375010602504486713997487440, 8.660280443238974622015930896966, 9.237873860136522866347651115197, 10.27659866254238548269165551282, 10.83948572200774822149232291464, 11.715625135551366099336108549283, 13.64099997932543745775506327367, 14.38093525352286638900733602014, 14.98541418822998325073697580306, 16.23312633449401316678434326063, 17.07495707911087514914813222444, 17.46151171821882455432930494649, 18.78047496552558646479185002944, 19.56515235555549551823901218375, 20.44368552759393186992468286029, 21.32955699308834074666492144642, 21.821632364991276152641980762627, 23.13652566079728179488959505225, 24.461034522007823571109607377678, 25.00326769947323590307058494112

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