Properties

Label 1-764-764.351-r1-0-0
Degree $1$
Conductor $764$
Sign $-0.890 + 0.454i$
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  + (−0.546 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ -0.890 + 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3800524654 + 1.582182761i\)
\(L(\frac12)\) \(\approx\) \(0.3800524654 + 1.582182761i\)
\(L(1)\) \(\approx\) \(0.8589653990 + 0.5026079897i\)
\(L(1)\) \(\approx\) \(0.8589653990 + 0.5026079897i\)

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

−22.04130474225533817816549141750, −21.0870110116093640080834917204, −20.19552670416375762606312573006, −19.31448609841698474175885700955, −18.42727989885633058862981546729, −17.99936517443272722949232567772, −16.95659260220913893461139161006, −16.36334214241804068349250490110, −15.62048411752121148807709502576, −14.01213610796039251742957422049, −13.55019997704517788095089722440, −12.93511104631713027644652002743, −12.114434110073756394064077128718, −11.08516617272897750608901739836, −10.26302749608074619676995895041, −9.32247690979990088391055447767, −8.41278588965196586627138661563, −7.33539776116954650914636702927, −6.42480952868761355348000688551, −5.69225537133241269142881325020, −5.133049739761579077643776238335, −3.320864236113034084013685466978, −2.57094169955254699238636375791, −1.16068318556112678827123026054, −0.46625526542785609447591620223, 1.17584888568250229683787668979, 2.557349748175933569414496519756, 3.50214406226045681500219185739, 4.59503461171393540783054623844, 5.46048045390262772075158223941, 6.52726473979840590644173470941, 6.82168458618451037930498892700, 8.63062487647361434767470367945, 9.445087534917486965911563058677, 10.08045067387457748479783505539, 10.67331696573910842911195309256, 11.767085034914763363498006635, 12.67693605910548363176504151681, 13.52373245379674567251441373354, 14.44609705196490867282644031791, 15.39130241649587895336896143246, 15.99048828930972307464811689975, 16.98285803989542666648556457570, 17.48252352161938437021649684539, 18.38247609540815845314862482388, 19.28164763993514982456356927986, 20.36051899563005271368378098908, 21.08815507996314318672147033976, 21.76537875321729013338131800090, 22.505021739302968077885846763800

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