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Properties

Label 1-763-763.9-r0-0-0
Degree $1$
Conductor $763$
Sign $-0.219 - 0.975i$
Analytic cond. $3.54335$
Root an. cond. $3.54335$
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  + (−0.939 − 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 + 0.642i)4-s + (0.597 − 0.802i)5-s + (0.973 − 0.230i)6-s + (−0.5 − 0.866i)8-s + (0.396 − 0.918i)9-s + (−0.835 + 0.549i)10-s + (−0.0581 + 0.998i)11-s + (−0.993 − 0.116i)12-s + (−0.993 − 0.116i)13-s + (−0.0581 + 0.998i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + (−0.686 + 0.727i)18-s + (0.766 − 0.642i)19-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.835 + 0.549i)3-s + (0.766 + 0.642i)4-s + (0.597 − 0.802i)5-s + (0.973 − 0.230i)6-s + (−0.5 − 0.866i)8-s + (0.396 − 0.918i)9-s + (−0.835 + 0.549i)10-s + (−0.0581 + 0.998i)11-s + (−0.993 − 0.116i)12-s + (−0.993 − 0.116i)13-s + (−0.0581 + 0.998i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + (−0.686 + 0.727i)18-s + (0.766 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(763\)    =    \(7 \cdot 109\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(3.54335\)
Root analytic conductor: \(3.54335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{763} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 763,\ (0:\ ),\ -0.219 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3343555675 - 0.4177204370i\)
\(L(\frac12)\) \(\approx\) \(0.3343555675 - 0.4177204370i\)
\(L(1)\) \(\approx\) \(0.5451590712 - 0.1271030002i\)
\(L(1)\) \(\approx\) \(0.5451590712 - 0.1271030002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.0581 + 0.998i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.0581 - 0.998i)T \)
31 \( 1 + (0.396 - 0.918i)T \)
37 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.973 + 0.230i)T \)
53 \( 1 + (-0.993 - 0.116i)T \)
59 \( 1 + (-0.835 - 0.549i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.0581 + 0.998i)T \)
79 \( 1 + (0.396 - 0.918i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (-0.686 - 0.727i)T \)
97 \( 1 + (0.597 - 0.802i)T \)
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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

−22.64194766569628000630584138845, −21.902546761681074698426003435053, −21.133441217092320994033169601904, −19.84210013456180198664049556947, −18.989704737845356979505613163598, −18.58777821727579982542767063780, −17.76814039919007270323447087507, −17.10816268878458935714028610271, −16.43183045263855756247873969795, −15.601266108929914759745837334376, −14.24756200102452257352770381333, −14.04249194815375672146789287887, −12.45798392798184761528431010592, −11.81898841775992840150783483184, −10.65955681045320882160857966257, −10.41971199606308157452105149769, −9.396742752163053823259345665992, −8.1576029974549381466831863010, −7.40753893326062840358868154992, −6.59726496183361633321462008872, −5.83733338412043677064359185055, −5.21347954550307643718699479655, −3.2721880176708880708123294732, −2.13092834392862508660085403573, −1.198405923402918612460097794061, 0.414219126310509644667056219175, 1.61113662009167057433382255232, 2.72535621259899123792864772729, 4.161081980154899796473390230403, 5.055572080941918250091068905135, 5.95218120778916772309318883432, 7.08711824454270254213309285027, 7.887603342623635281830206032731, 9.23644329536583927956499347800, 9.772134150863137177369950254264, 10.15120310047809850187240227392, 11.463102256760153683790322625646, 12.114146043352822325055792519888, 12.65693792095339273583009437092, 13.920522864366625165881659981, 15.31547269661061916469280073263, 15.850473651137775731601233197703, 16.83792408377967037729448612282, 17.308160633560643401052642268324, 17.86258702204101324148168291254, 18.76907222537563577414797989214, 20.009355953341824387185495929070, 20.4997040893725238507244114037, 21.2178326687162963946666828088, 22.03011002737974626117364581736

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