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Properties

Label 1-764-764.759-r0-0-0
Degree $1$
Conductor $764$
Sign $0.487 + 0.873i$
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  + (0.986 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.487 + 0.873i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ 0.487 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.084796454 + 1.223696955i\)
\(L(\frac12)\) \(\approx\) \(2.084796454 + 1.223696955i\)
\(L(1)\) \(\approx\) \(1.616146116 + 0.4669554504i\)
\(L(1)\) \(\approx\) \(1.616146116 + 0.4669554504i\)

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

−21.75303040446293194389904401451, −21.314221622592909114348151212433, −20.84594990507389227717852780421, −19.703787951682862348130595605181, −19.38136114972206926548099145832, −18.1118908757772295263015337645, −17.51328976893299837990397784813, −16.684799225928077983492878546545, −15.49549607865345872013101092298, −14.97176611851012779946278081083, −13.86578547101953903085889794162, −13.4571987260436992973273893651, −12.57046379853066221972213772740, −11.64638338178122596284561047891, −10.43816213279454767507622838313, −9.65611504230261905752964923214, −8.68641609412255265950557083340, −8.13612103282343371331457346481, −7.37222192114654660299516287297, −5.98744182847838313961575566420, −5.01565280815360246376147234064, −4.26080256969937850264241504528, −2.895301577980724160888958398814, −2.047541474631055990602675748543, −1.04934407738390951970317137460, 1.68308146727817675621361796317, 2.39964198230468508822206247826, 3.2145423067415158230175972200, 4.58661146082527174276650972295, 5.18319777203667199137004266989, 6.71924028636168373317181246766, 7.45442028203394737559537841471, 8.16407028721929708254797164966, 9.24169429052498100070227596677, 10.16662428294903726936594898389, 10.55089910200747520558448166641, 11.89358166411946497664820294261, 12.73523240566478565307437335705, 13.96317064797534065159782028435, 14.30682886484734172596089989136, 14.9550751823039123126256687059, 15.79093976559193963196985732350, 16.95108938963345515029798894159, 17.94093640686676800647891003263, 18.45975933444197160010667061028, 19.24658901428451821927731929329, 20.35370094098609408265759795899, 20.88502563662362343039080807825, 21.515802566927379361330704847484, 22.44562534432889124980934147591

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