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Properties

Label 1-764-764.243-r1-0-0
Degree $1$
Conductor $764$
Sign $-0.577 + 0.816i$
Analytic cond. $82.1032$
Root an. cond. $82.1032$
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  + (−0.789 − 0.614i)3-s + (−0.677 − 0.735i)5-s − 7-s + (0.245 + 0.969i)9-s + (0.986 − 0.164i)11-s + (0.789 − 0.614i)13-s + (0.0825 + 0.996i)15-s + (0.546 + 0.837i)17-s + (−0.945 − 0.324i)19-s + (0.789 + 0.614i)21-s + (−0.945 − 0.324i)23-s + (−0.0825 + 0.996i)25-s + (0.401 − 0.915i)27-s + (−0.0825 − 0.996i)29-s + (−0.245 − 0.969i)31-s + ⋯
L(s)  = 1  + (−0.789 − 0.614i)3-s + (−0.677 − 0.735i)5-s − 7-s + (0.245 + 0.969i)9-s + (0.986 − 0.164i)11-s + (0.789 − 0.614i)13-s + (0.0825 + 0.996i)15-s + (0.546 + 0.837i)17-s + (−0.945 − 0.324i)19-s + (0.789 + 0.614i)21-s + (−0.945 − 0.324i)23-s + (−0.0825 + 0.996i)25-s + (0.401 − 0.915i)27-s + (−0.0825 − 0.996i)29-s + (−0.245 − 0.969i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1655734047 - 0.3199624553i\)
\(L(\frac12)\) \(\approx\) \(-0.1655734047 - 0.3199624553i\)
\(L(1)\) \(\approx\) \(0.5672212313 - 0.2815507259i\)
\(L(1)\) \(\approx\) \(0.5672212313 - 0.2815507259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
191 \( 1 \)
good3 \( 1 + (-0.789 - 0.614i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 - T \)
11 \( 1 + (0.986 - 0.164i)T \)
13 \( 1 + (0.789 - 0.614i)T \)
17 \( 1 + (0.546 + 0.837i)T \)
19 \( 1 + (-0.945 - 0.324i)T \)
23 \( 1 + (-0.945 - 0.324i)T \)
29 \( 1 + (-0.0825 - 0.996i)T \)
31 \( 1 + (-0.245 - 0.969i)T \)
37 \( 1 + (0.245 - 0.969i)T \)
41 \( 1 + (-0.401 - 0.915i)T \)
43 \( 1 + (0.879 - 0.475i)T \)
47 \( 1 + (0.986 - 0.164i)T \)
53 \( 1 + (-0.986 + 0.164i)T \)
59 \( 1 + (-0.245 - 0.969i)T \)
61 \( 1 + (0.546 - 0.837i)T \)
67 \( 1 + (-0.546 + 0.837i)T \)
71 \( 1 + (0.401 + 0.915i)T \)
73 \( 1 + (-0.986 - 0.164i)T \)
79 \( 1 + (0.0825 - 0.996i)T \)
83 \( 1 + (-0.945 + 0.324i)T \)
89 \( 1 + (-0.879 - 0.475i)T \)
97 \( 1 + (0.245 - 0.969i)T \)
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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

−22.63163987114338491642136420993, −22.13731766020558062455671574252, −21.340352100641715490950807869874, −20.25553360982410444729758814507, −19.47095371504619342986989403594, −18.63676031163413890626868485471, −17.969063406499892595205792209768, −16.75801775178896340642456721227, −16.27792090272932686330283660997, −15.58826411757469522163212636568, −14.67874295621142734565614274469, −13.901700972504368637536056714114, −12.53532007710856927975769710354, −11.92151869591240837699235766139, −11.16523583317167171911359437958, −10.33343350174797390710404177313, −9.554071875842042240497994710888, −8.68160335511801742939165515894, −7.24965753176981096441559571023, −6.51705424470176177057157019992, −5.95053542290335994860798620856, −4.49807783753640563830909548043, −3.77867436055411751485470433358, −3.040651116295134309438455001946, −1.27019386256991678336228750205, 0.129736862895921503896843898834, 0.843393623520154495604308528343, 2.07874512260945443464460575267, 3.69119067957666000848214378799, 4.27050102980790230531083539894, 5.80476295121129362154006266049, 6.11142004752172079817906034316, 7.24730236085888301887173248987, 8.15609174730596393229461840081, 8.97522570817620648930014136757, 10.129038621557025565080299111690, 11.04374631284060978676658522107, 11.89237766882677265729998234485, 12.66946139235470121609590423617, 13.0533011147832674020605660256, 14.15961578452147065406573536481, 15.49157528391417343706626780447, 16.0301149274979611288687685516, 17.01197505989368595312286461567, 17.272048562812629547519105927806, 18.68380566134383963295513553146, 19.18449568772307218272752401305, 19.85437516687235339505715600715, 20.77426571424846571615328134990, 21.989717433794732480525477488559

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