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Properties

Label 1-764-764.559-r1-0-0
Degree $1$
Conductor $764$
Sign $-0.153 - 0.988i$
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  + (0.0825 − 0.996i)3-s + (−0.879 − 0.475i)5-s − 7-s + (−0.986 − 0.164i)9-s + (0.401 + 0.915i)11-s + (−0.0825 − 0.996i)13-s + (−0.546 + 0.837i)15-s + (0.789 − 0.614i)17-s + (0.677 + 0.735i)19-s + (−0.0825 + 0.996i)21-s + (0.677 + 0.735i)23-s + (0.546 + 0.837i)25-s + (−0.245 + 0.969i)27-s + (0.546 − 0.837i)29-s + (0.986 + 0.164i)31-s + ⋯
L(s)  = 1  + (0.0825 − 0.996i)3-s + (−0.879 − 0.475i)5-s − 7-s + (−0.986 − 0.164i)9-s + (0.401 + 0.915i)11-s + (−0.0825 − 0.996i)13-s + (−0.546 + 0.837i)15-s + (0.789 − 0.614i)17-s + (0.677 + 0.735i)19-s + (−0.0825 + 0.996i)21-s + (0.677 + 0.735i)23-s + (0.546 + 0.837i)25-s + (−0.245 + 0.969i)27-s + (0.546 − 0.837i)29-s + (0.986 + 0.164i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $-0.153 - 0.988i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ -0.153 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8802480659 - 1.027913217i\)
\(L(\frac12)\) \(\approx\) \(0.8802480659 - 1.027913217i\)
\(L(1)\) \(\approx\) \(0.8028913746 - 0.3529181969i\)
\(L(1)\) \(\approx\) \(0.8028913746 - 0.3529181969i\)

Euler product

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

−22.22906654950390578859243801506, −21.838190494314157031330114281797, −20.87325219544689264939047747090, −19.88686491761261266703336397451, −19.21518674923557994935517234418, −18.78943848962340528089241084358, −17.2932116496292176830957953715, −16.41704305536236321024133561380, −16.05283053476902352233444018469, −15.190481329001871131910765502068, −14.35504353517528855114914228652, −13.65991793541703978707755363670, −12.29370185769408870794957965308, −11.60513619665509598313741454940, −10.75039571309830460458025968227, −10.02887529345596505842134978941, −8.994466473364009099656640409797, −8.43831464525356623087387684479, −7.07649241920662334266216715326, −6.36493543913230078668426481770, −5.21014805409863838168804315853, −4.09965427949355540035746727149, −3.421047568786243856693779174927, −2.73427222771876643064702536333, −0.710729923458167599912376344910, 0.497536124626587132115516391405, 1.35547359723132706195005278323, 2.88852920092861240072038092320, 3.50792687399841107790779036285, 4.889783841593817467799341711086, 5.86084171132161209756389154109, 6.93831536637334193868186970392, 7.57238418033332293234532279981, 8.32256287327813253416763229822, 9.41398137455530988255969187649, 10.2032231748777503634903542791, 11.71881087525630306757207360528, 12.069152325987307921425361952287, 12.855417909501118766055376792028, 13.53675001525652617079099370480, 14.64725676611811161603332885602, 15.48990434503781048888424907963, 16.29483511147709378242382464804, 17.2079308880562290127939832289, 17.942496201626252050439254037337, 19.07404281376525981568447748089, 19.36483439751637119711608597418, 20.266828787710384432506672308044, 20.77907096149392501444816912279, 22.44037633833263187046989323659

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