Properties

Label 1-764-764.223-r1-0-0
Degree $1$
Conductor $764$
Sign $-0.609 + 0.792i$
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  + (0.401 − 0.915i)3-s + (0.789 − 0.614i)5-s − 7-s + (−0.677 − 0.735i)9-s + (0.879 − 0.475i)11-s + (−0.401 − 0.915i)13-s + (−0.245 − 0.969i)15-s + (−0.986 + 0.164i)17-s + (−0.546 − 0.837i)19-s + (−0.401 + 0.915i)21-s + (−0.546 − 0.837i)23-s + (0.245 − 0.969i)25-s + (−0.945 + 0.324i)27-s + (0.245 + 0.969i)29-s + (0.677 + 0.735i)31-s + ⋯
L(s)  = 1  + (0.401 − 0.915i)3-s + (0.789 − 0.614i)5-s − 7-s + (−0.677 − 0.735i)9-s + (0.879 − 0.475i)11-s + (−0.401 − 0.915i)13-s + (−0.245 − 0.969i)15-s + (−0.986 + 0.164i)17-s + (−0.546 − 0.837i)19-s + (−0.401 + 0.915i)21-s + (−0.546 − 0.837i)23-s + (0.245 − 0.969i)25-s + (−0.945 + 0.324i)27-s + (0.245 + 0.969i)29-s + (0.677 + 0.735i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $-0.609 + 0.792i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ -0.609 + 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4538716254 - 0.9217324137i\)
\(L(\frac12)\) \(\approx\) \(-0.4538716254 - 0.9217324137i\)
\(L(1)\) \(\approx\) \(0.8518860501 - 0.6153477240i\)
\(L(1)\) \(\approx\) \(0.8518860501 - 0.6153477240i\)

Euler product

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

−22.49152045043627560453603022197, −21.94895721764521531710365300341, −21.24074281575453755098170007665, −20.38458543496169871947588893887, −19.347328624761035473190809958067, −19.07774890140112867166898002142, −17.59722412919455598166757257463, −17.09396995672314892298422899099, −16.13541677935373658200685943838, −15.40693552390747287800676503385, −14.46470283146201366626926383206, −13.96181935497108938040258847319, −13.066509542573545276819765725773, −11.88566931013602357673687503923, −10.976990200530877162876123723356, −9.93493539225246731818010637885, −9.58249178923841492776629358250, −8.8754718190327413598202960268, −7.48433634308214943629141176022, −6.45782810878950382836495306718, −5.8642916507737307262075692701, −4.43373795704037588609067152543, −3.80338623372024093244340915402, −2.64093700405861230084050444895, −1.89442993807719097904929762061, 0.21219226264160080355288724731, 1.14258135807926102711046653675, 2.34399963205928880517600572772, 3.101446031764931382706075290164, 4.407689993308853647050207050813, 5.70767409658566831978920388949, 6.44816286553310964894025631697, 7.05509287261719954908930950297, 8.57795398419768204881226526289, 8.81380715349636324260398101176, 9.8533069362147259113160233619, 10.83024706542748819746367471861, 12.23831612242570141562176521813, 12.60438508625938669461161391073, 13.48108988562396814890312959725, 13.990424884614366865882993206569, 15.0677538504328198827873075078, 16.04790342649814597767207450262, 17.08199447528579585251462929112, 17.54989856361171648083446039494, 18.44852902315973907967399719513, 19.49715116795801013671074285924, 19.86983120376017956782680601171, 20.622468386209711436265342833517, 21.92134769246687750135004566892

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