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Properties

Label 1-764-764.695-r0-0-0
Degree $1$
Conductor $764$
Sign $0.978 + 0.204i$
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  + (0.677 + 0.735i)3-s + (0.245 − 0.969i)5-s + 7-s + (−0.0825 + 0.996i)9-s + (0.546 − 0.837i)11-s + (−0.677 + 0.735i)13-s + (0.879 − 0.475i)15-s + (0.945 − 0.324i)17-s + (−0.401 + 0.915i)19-s + (0.677 + 0.735i)21-s + (0.401 − 0.915i)23-s + (−0.879 − 0.475i)25-s + (−0.789 + 0.614i)27-s + (0.879 − 0.475i)29-s + (−0.0825 + 0.996i)31-s + ⋯
L(s)  = 1  + (0.677 + 0.735i)3-s + (0.245 − 0.969i)5-s + 7-s + (−0.0825 + 0.996i)9-s + (0.546 − 0.837i)11-s + (−0.677 + 0.735i)13-s + (0.879 − 0.475i)15-s + (0.945 − 0.324i)17-s + (−0.401 + 0.915i)19-s + (0.677 + 0.735i)21-s + (0.401 − 0.915i)23-s + (−0.879 − 0.475i)25-s + (−0.789 + 0.614i)27-s + (0.879 − 0.475i)29-s + (−0.0825 + 0.996i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (695, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ 0.978 + 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.204577977 + 0.2274938749i\)
\(L(\frac12)\) \(\approx\) \(2.204577977 + 0.2274938749i\)
\(L(1)\) \(\approx\) \(1.546316152 + 0.1332538048i\)
\(L(1)\) \(\approx\) \(1.546316152 + 0.1332538048i\)

Euler product

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

−22.391665202529292971779138411274, −21.47603760080098926291476350648, −20.75337935534045019230390646877, −19.74962252068272652299413589106, −19.27385871004689788951244491412, −18.23896888591791185868164112887, −17.64228008875140394383216794898, −17.178978237005966374993551048614, −15.380282936851443042367751775485, −14.84984656433492797229135404993, −14.36022456828642568860646427304, −13.48753197435457220316253083831, −12.50662437135677896873363367784, −11.74255517316846745242070594884, −10.78431423066146874564664837714, −9.820525725379942128527612453896, −8.97384620429911325387172292399, −7.714073521504831147859414604319, −7.44849719899547209949965415843, −6.4438705909389057837062771980, −5.37762018065492842548554965089, −4.13842941496798703380940004494, −2.99556004497637176103992354094, −2.21749974925095316041755143161, −1.24646901236915758938003065575, 1.21150939484780853087712638543, 2.1994306807533943834553981156, 3.45332124204223342948994130534, 4.517937839569467728365060043708, 4.99165025075184235179799926878, 6.09019861329083654353558035866, 7.55854715270680072683250717049, 8.446368215981090324978083891855, 8.86938475155083399436258144568, 9.85714989152514564965257655648, 10.6625796139045749582785034020, 11.7850204857348986527301476806, 12.42704905587381829496253230593, 13.903983004247551108985177102478, 14.07787585091240617172289586768, 14.968573313355221335213722431855, 16.04221546767392857634574956410, 16.80091362652646746682893136249, 17.15479162222358947240371840486, 18.60313482622844966138312451897, 19.2925312513673750761743148403, 20.23294201950457586733253434724, 20.91673769335849749008241419090, 21.38361539359415197393396260141, 22.07632978748346644339490521052

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