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Properties

Label 1-764-764.31-r0-0-0
Degree $1$
Conductor $764$
Sign $0.492 + 0.870i$
Analytic cond. $3.54800$
Root an. cond. $3.54800$
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  + (−0.546 + 0.837i)3-s + (0.945 + 0.324i)5-s + 7-s + (−0.401 − 0.915i)9-s + (0.245 − 0.969i)11-s + (0.546 + 0.837i)13-s + (−0.789 + 0.614i)15-s + (−0.0825 + 0.996i)17-s + (−0.879 + 0.475i)19-s + (−0.546 + 0.837i)21-s + (0.879 − 0.475i)23-s + (0.789 + 0.614i)25-s + (0.986 + 0.164i)27-s + (−0.789 + 0.614i)29-s + (−0.401 − 0.915i)31-s + ⋯
L(s)  = 1  + (−0.546 + 0.837i)3-s + (0.945 + 0.324i)5-s + 7-s + (−0.401 − 0.915i)9-s + (0.245 − 0.969i)11-s + (0.546 + 0.837i)13-s + (−0.789 + 0.614i)15-s + (−0.0825 + 0.996i)17-s + (−0.879 + 0.475i)19-s + (−0.546 + 0.837i)21-s + (0.879 − 0.475i)23-s + (0.789 + 0.614i)25-s + (0.986 + 0.164i)27-s + (−0.789 + 0.614i)29-s + (−0.401 − 0.915i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.427855841 + 0.8326830377i\)
\(L(\frac12)\) \(\approx\) \(1.427855841 + 0.8326830377i\)
\(L(1)\) \(\approx\) \(1.149640675 + 0.3750248596i\)
\(L(1)\) \(\approx\) \(1.149640675 + 0.3750248596i\)

Euler product

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

−22.42366177501763990210502270073, −21.39616331233302547857059528911, −20.67127705707347805810263795721, −19.961224919761035379792640462801, −18.79746882178311908686053226677, −17.96418539952611468963425285321, −17.53889049681616996288304583255, −17.01223852826991112513816158665, −15.82910656704325686696986563955, −14.73982397047410869643422159566, −13.98532019588916257891714173808, −13.03165573685459099923993113548, −12.64294676673286360243620729250, −11.41899453541560504098200566420, −10.90429485722722725937669257931, −9.77060419509821040770428861357, −8.807996646104935529021113592000, −7.84706912027728820223242689917, −7.01686456908635510966683301551, −6.076360290956395771511194606416, −5.18195066043925321320137435544, −4.58778549472482140453557320787, −2.72668030397952425220257173435, −1.81537090459886091399157885330, −1.00202151779673338193198137471, 1.22915795304678139086655809048, 2.32548187419658127070278408041, 3.73455447474185116627678295099, 4.43478214045375092029699300535, 5.782600101947651762273789950563, 5.93296024412356268108135141159, 7.19464515654114146260948184193, 8.77983189671543181733121906602, 8.95709584043434284620353574773, 10.34997963468951395342497095103, 10.8885011224401121388588601633, 11.4349533083183649061186804155, 12.65600617663241444434008045024, 13.67663260553671592989119941526, 14.64551536240420041651853541191, 14.89192939894318389630042466634, 16.3133358369276190767624992871, 16.87557765006522763309451135142, 17.49407203664805509618315342661, 18.41188346484778173973090412568, 19.11612767348076474268134299150, 20.55900991036593231232590360397, 21.13441766414121816557481955781, 21.625351706376173636501933976100, 22.25855984578952429387430172534

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