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Properties

Label 1-367-367.106-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.917 + 0.396i$
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  + (−0.376 + 0.926i)2-s + (0.952 + 0.304i)3-s + (−0.716 − 0.697i)4-s + (−0.279 + 0.960i)5-s + (−0.640 + 0.767i)6-s + (−0.469 + 0.882i)7-s + (0.916 − 0.400i)8-s + (0.815 + 0.579i)9-s + (−0.784 − 0.620i)10-s + (0.229 − 0.973i)11-s + (−0.469 − 0.882i)12-s + (0.815 + 0.579i)13-s + (−0.640 − 0.767i)14-s + (−0.558 + 0.829i)15-s + (0.0257 + 0.999i)16-s + (0.128 + 0.991i)17-s + ⋯
L(s)  = 1  + (−0.376 + 0.926i)2-s + (0.952 + 0.304i)3-s + (−0.716 − 0.697i)4-s + (−0.279 + 0.960i)5-s + (−0.640 + 0.767i)6-s + (−0.469 + 0.882i)7-s + (0.916 − 0.400i)8-s + (0.815 + 0.579i)9-s + (−0.784 − 0.620i)10-s + (0.229 − 0.973i)11-s + (−0.469 − 0.882i)12-s + (0.815 + 0.579i)13-s + (−0.640 − 0.767i)14-s + (−0.558 + 0.829i)15-s + (0.0257 + 0.999i)16-s + (0.128 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.917 + 0.396i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.917 + 0.396i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2507639949 + 1.212738342i\)
\(L(\frac12)\) \(\approx\) \(0.2507639949 + 1.212738342i\)
\(L(1)\) \(\approx\) \(0.7285476954 + 0.7773405557i\)
\(L(1)\) \(\approx\) \(0.7285476954 + 0.7773405557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.376 + 0.926i)T \)
3 \( 1 + (0.952 + 0.304i)T \)
5 \( 1 + (-0.279 + 0.960i)T \)
7 \( 1 + (-0.469 + 0.882i)T \)
11 \( 1 + (0.229 - 0.973i)T \)
13 \( 1 + (0.815 + 0.579i)T \)
17 \( 1 + (0.128 + 0.991i)T \)
19 \( 1 + (0.423 + 0.905i)T \)
23 \( 1 + (0.0257 - 0.999i)T \)
29 \( 1 + (-0.998 - 0.0514i)T \)
31 \( 1 + (-0.716 - 0.697i)T \)
37 \( 1 + (-0.935 + 0.352i)T \)
41 \( 1 + (0.229 + 0.973i)T \)
43 \( 1 + (0.423 - 0.905i)T \)
47 \( 1 + (0.514 + 0.857i)T \)
53 \( 1 + (-0.279 + 0.960i)T \)
59 \( 1 + (-0.0771 - 0.997i)T \)
61 \( 1 + (-0.640 - 0.767i)T \)
67 \( 1 + (-0.0771 + 0.997i)T \)
71 \( 1 + (-0.716 + 0.697i)T \)
73 \( 1 + (0.994 + 0.102i)T \)
79 \( 1 + (-0.558 - 0.829i)T \)
83 \( 1 + T \)
89 \( 1 + (0.514 + 0.857i)T \)
97 \( 1 + (0.0257 - 0.999i)T \)
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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

−24.33198810575596434550421333162, −23.30580565102900093702680155853, −22.56724025794101822974792361863, −21.121571518133085337680342662511, −20.52414311058344288101198448885, −19.927991443620727629336186969975, −19.45465942168542581350175268148, −18.18463216230120507328833258761, −17.49917428361995881438703566752, −16.34157341347212414299593891227, −15.46232159733152118836821071224, −13.974990508278594313807739886299, −13.29976289289989468778030356714, −12.69096732571402848429626866195, −11.72013968953960023827213297111, −10.475123239600140917750097713697, −9.36484705368578747591984684245, −9.03231542121651950208876462794, −7.68984179188815320749487903519, −7.20464091149144063548457226585, −5.070764361023720772139346007472, −3.95440267398437052548384560242, −3.271446713490825193930352679642, −1.816488807788094086260054325575, −0.8153637920797272986217190685, 1.83172237824296577919260706281, 3.29106019560056878804866494801, 4.055850359639792930076758507530, 5.78108169493404670668685684458, 6.44379207327601467660457642311, 7.65822254716118753013274654623, 8.49503191465190799156608750240, 9.2026599055981437719585708572, 10.23569937352676415478251599567, 11.14424352083230464639923454718, 12.73414847230734938642704608805, 13.86855072167346462173846463361, 14.48128500540958724311319121809, 15.26413578254268941742955367949, 15.993636390997836220597593562284, 16.767819022165089790176437242842, 18.39723956619518808444906156986, 18.835198364699798445138708406095, 19.256046849776377780068162069890, 20.60730917098677411910726246975, 21.882992144576514473864809638194, 22.28279797189394735937852068786, 23.48921904603210840090466049240, 24.43984179708087719284281475780, 25.18580355472731177261749324904

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