Properties

Label 1-764-764.503-r1-0-0
Degree $1$
Conductor $764$
Sign $0.906 + 0.423i$
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  + (−0.245 + 0.969i)3-s + (−0.0825 − 0.996i)5-s − 7-s + (−0.879 − 0.475i)9-s + (−0.945 − 0.324i)11-s + (0.245 + 0.969i)13-s + (0.986 + 0.164i)15-s + (−0.401 − 0.915i)17-s + (−0.789 + 0.614i)19-s + (0.245 − 0.969i)21-s + (−0.789 + 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.677 − 0.735i)27-s + (−0.986 − 0.164i)29-s + (0.879 + 0.475i)31-s + ⋯
L(s)  = 1  + (−0.245 + 0.969i)3-s + (−0.0825 − 0.996i)5-s − 7-s + (−0.879 − 0.475i)9-s + (−0.945 − 0.324i)11-s + (0.245 + 0.969i)13-s + (0.986 + 0.164i)15-s + (−0.401 − 0.915i)17-s + (−0.789 + 0.614i)19-s + (0.245 − 0.969i)21-s + (−0.789 + 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.677 − 0.735i)27-s + (−0.986 − 0.164i)29-s + (0.879 + 0.475i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.906 + 0.423i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ 0.906 + 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5866528122 + 0.1302544635i\)
\(L(\frac12)\) \(\approx\) \(0.5866528122 + 0.1302544635i\)
\(L(1)\) \(\approx\) \(0.6322123641 + 0.08499321519i\)
\(L(1)\) \(\approx\) \(0.6322123641 + 0.08499321519i\)

Euler product

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

−22.45815421912884235541896038522, −21.53751781624434316909144872383, −20.264656365104010143117052943511, −19.52976665684998228892777518511, −18.89612723814516722701088681029, −18.129352899410215234149206022855, −17.57879708128512843935403436119, −16.54173731462967640988017121573, −15.471160836190786212388270708272, −14.91583110326781049120939890230, −13.71066790660248254809151318764, −13.00515226935608444185258992632, −12.5369189971939165904541542136, −11.30949019922714580230388701263, −10.58650495848159664417751137531, −9.892579039019639799007151321745, −8.400400769045511683129526231401, −7.78074575220606973088244002419, −6.679466888119019219301639999066, −6.31601466624132219002545742986, −5.28240887540702728741024459905, −3.74003242812612028693636254601, −2.75923885620466627813642217561, −2.0623908276066149402809503253, −0.35011366315742783832157164782, 0.37484192619437549203894439095, 2.06972394923256410115646006847, 3.39229592889267064552379433094, 4.14538161179191386803108541264, 5.12149655720524937735510243928, 5.8421465976781099690378657273, 6.88876048399952094087966932247, 8.28959484080244939582190240778, 8.95936876606892612515127546489, 9.76480593472813830660050103778, 10.44140725090237758785517281440, 11.60350794671162527215030928204, 12.2116375043995501244863077598, 13.3452572955502791581219061240, 13.88527602183531616112603248729, 15.343436616391153287283791573417, 15.81782996064752447316297109119, 16.531488851947565985310306087601, 16.99012975968880984859298744646, 18.20174680811404234596846249363, 19.16006236729189362168373732409, 19.988940333693946528833397523707, 20.83621751738548472089049808428, 21.28016913034482996569406640568, 22.20191145243290482670803854255

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