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Properties

Label 1-764-764.11-r0-0-0
Degree $1$
Conductor $764$
Sign $0.952 + 0.303i$
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  + (−0.789 + 0.614i)3-s + (−0.677 + 0.735i)5-s + 7-s + (0.245 − 0.969i)9-s + (−0.986 − 0.164i)11-s + (0.789 + 0.614i)13-s + (0.0825 − 0.996i)15-s + (0.546 − 0.837i)17-s + (0.945 − 0.324i)19-s + (−0.789 + 0.614i)21-s + (−0.945 + 0.324i)23-s + (−0.0825 − 0.996i)25-s + (0.401 + 0.915i)27-s + (0.0825 − 0.996i)29-s + (0.245 − 0.969i)31-s + ⋯
L(s)  = 1  + (−0.789 + 0.614i)3-s + (−0.677 + 0.735i)5-s + 7-s + (0.245 − 0.969i)9-s + (−0.986 − 0.164i)11-s + (0.789 + 0.614i)13-s + (0.0825 − 0.996i)15-s + (0.546 − 0.837i)17-s + (0.945 − 0.324i)19-s + (−0.789 + 0.614i)21-s + (−0.945 + 0.324i)23-s + (−0.0825 − 0.996i)25-s + (0.401 + 0.915i)27-s + (0.0825 − 0.996i)29-s + (0.245 − 0.969i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ 0.952 + 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.007582145 + 0.1567376970i\)
\(L(\frac12)\) \(\approx\) \(1.007582145 + 0.1567376970i\)
\(L(1)\) \(\approx\) \(0.8296721169 + 0.1638671862i\)
\(L(1)\) \(\approx\) \(0.8296721169 + 0.1638671862i\)

Euler product

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

−22.57622879790545809655419946879, −21.48285653593490725805916947534, −20.716306036020329450689692550190, −20.018585778695240343877937625399, −19.00619938022236322420180557765, −18.06307226635201803410359582204, −17.79575184706306383022413763429, −16.58969680167089187935424110143, −16.06787916716298806836199866345, −15.18331932494175729382009114941, −14.04077660199722938843742784198, −13.12615288652694604658265713774, −12.38310396916623034852056912611, −11.77138757025163648166698179933, −10.847685465322958865677612449518, −10.21029078648538418172867580046, −8.552655866344565803603095475529, −8.00557932955709834722992477309, −7.40884607511597881540802990930, −6.0185595222966000272494855088, −5.2491007883603303631336458307, −4.59059201237753207650073282866, −3.303109500518245984884922304921, −1.73414837746641253758698168300, −0.96701088368267796563284474165, 0.72192357547582220418784275506, 2.35349877223880208242419176178, 3.58100994067564023888318499297, 4.37166818172617417209361565637, 5.337725125829274377122449397319, 6.13585025178458069566723732464, 7.37615356320561679505006828670, 7.935740002142972631111858695222, 9.203687711828786639295150848278, 10.190873718588795836722412855457, 11.006679649277300298406712979728, 11.53369255880278408962182163020, 12.13531276326501818261323951157, 13.63224302040433422740581928914, 14.31932557644989014084715082654, 15.39336727476183239921402919210, 15.830406732558411046375701753966, 16.59990423134054455994392026472, 17.8773437419942183953397732133, 18.15953708654438967146366577235, 18.98843218233815354233207642235, 20.25308327389094434565779799105, 21.03921212185356427969868227237, 21.50305450193633093659583086019, 22.67132806966010690128915667605

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