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Properties

Label 1-764-764.535-r1-0-0
Degree $1$
Conductor $764$
Sign $0.368 + 0.929i$
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  + (0.986 − 0.164i)3-s + (0.546 − 0.837i)5-s − 7-s + (0.945 − 0.324i)9-s + (0.677 + 0.735i)11-s + (−0.986 − 0.164i)13-s + (0.401 − 0.915i)15-s + (0.245 + 0.969i)17-s + (0.0825 + 0.996i)19-s + (−0.986 + 0.164i)21-s + (0.0825 + 0.996i)23-s + (−0.401 − 0.915i)25-s + (0.879 − 0.475i)27-s + (−0.401 + 0.915i)29-s + (−0.945 + 0.324i)31-s + ⋯
L(s)  = 1  + (0.986 − 0.164i)3-s + (0.546 − 0.837i)5-s − 7-s + (0.945 − 0.324i)9-s + (0.677 + 0.735i)11-s + (−0.986 − 0.164i)13-s + (0.401 − 0.915i)15-s + (0.245 + 0.969i)17-s + (0.0825 + 0.996i)19-s + (−0.986 + 0.164i)21-s + (0.0825 + 0.996i)23-s + (−0.401 − 0.915i)25-s + (0.879 − 0.475i)27-s + (−0.401 + 0.915i)29-s + (−0.945 + 0.324i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.368 + 0.929i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ 0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.926073252 + 1.308002615i\)
\(L(\frac12)\) \(\approx\) \(1.926073252 + 1.308002615i\)
\(L(1)\) \(\approx\) \(1.403014656 + 0.04286695483i\)
\(L(1)\) \(\approx\) \(1.403014656 + 0.04286695483i\)

Euler product

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

−22.09728166049873814312651166983, −21.39433956535536144250925367423, −20.24367714305717021483945300029, −19.6652869928287726485755688967, −18.81368233164520381639045604931, −18.39435794073949934690348730787, −17.0151600754503801811474015326, −16.38663883511519164622777267830, −15.285352442537725841635108926654, −14.70494266848658123631829602919, −13.78864511800661067844766853869, −13.38408993869218467049860034262, −12.2538772985256790229066329967, −11.154046827268466839171545646500, −10.1452681686460871091433575439, −9.465049058416567309319751659735, −8.94782432346446618437620731155, −7.54454870479426866258194878896, −6.90677017033960315749336860256, −6.056835182032592801494252198712, −4.73741331842398487798485362592, −3.543512997908455848324944502617, −2.85465027298772264120400329980, −2.11614937768608923019902325444, −0.4278305822592047289521978041, 1.29877719929110630059828215904, 2.01343123211109947428087484108, 3.274074374560612976361879171085, 4.06235835666598285258891129599, 5.220524016569384408409742907861, 6.29885000610026055401675601783, 7.2373730589221898718786179131, 8.123587071665774234134179589639, 9.182880236594619336861889868548, 9.63082008589975367486555858508, 10.29642769235158317510128478236, 12.05514850949210174202796823435, 12.68887294328506494418744595626, 13.14044839274786854918906593028, 14.26390650450980573810152971275, 14.832423167044622587617403529126, 15.82562261926412762420195904557, 16.73260182473997917107656558476, 17.36671268991293451902245213642, 18.46914357034286607884130036217, 19.36182406138116410111063219985, 20.0125560616297505438540514793, 20.4373508571962205502645498234, 21.64493471587756075697616945958, 22.02208897988892710890781896413

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