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Properties

Label 1-760-760.653-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.277 + 0.960i$
Analytic cond. $81.6733$
Root an. cond. $81.6733$
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  + (0.866 + 0.5i)3-s − i·7-s + (0.5 + 0.866i)9-s − 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.5 − 0.866i)29-s + 31-s + (−0.866 − 0.5i)33-s − i·37-s − 39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s − i·7-s + (0.5 + 0.866i)9-s − 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.5 − 0.866i)29-s + 31-s + (−0.866 − 0.5i)33-s − i·37-s − 39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.277 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.200722709 + 1.595930472i\)
\(L(\frac12)\) \(\approx\) \(1.200722709 + 1.595930472i\)
\(L(1)\) \(\approx\) \(1.240580343 + 0.3027254113i\)
\(L(1)\) \(\approx\) \(1.240580343 + 0.3027254113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−21.81386383654984557793149836525, −21.03460463277668755926391157087, −20.45680605210619699487689830957, −19.355666153132614044267651596467, −18.86875829107009889304563232709, −18.12011456205094628612284498492, −17.33650889352618504873041617261, −16.01069949454811101484123551630, −15.3522924080870332221588024830, −14.64138549531333231062654975879, −13.83200363718616916684686714485, −12.712973066562831297755317557400, −12.46525229033199896506399475623, −11.319395102434398252093727595064, −10.074557383545649428959274854442, −9.37049663202854322124111180111, −8.4587028044616497907810821473, −7.69824365204344645710239406188, −6.95305346528506333746603774775, −5.63246606390315533255782078594, −4.962271193430562212332265141186, −3.36118393042038155293492321592, −2.73216448786843693509378207903, −1.833554493411624165000373028662, −0.39743628650775769226510391989, 1.17922822501620431759927976442, 2.48346437954756947158192840459, 3.28636927619505565052359224318, 4.40440407116437081582472564528, 4.980094633859345953743402380475, 6.43522212883918696264740195507, 7.63110901581602765941733828684, 7.93903463007944535195341814886, 9.167622527455673524689133155921, 10.07920045433603140153940013234, 10.4741044313210733951262107803, 11.62423687750792522244756718219, 12.89766969081935751125207703733, 13.44802432842891267455693972275, 14.39875254909777814986369478852, 14.94760911664404339600085590791, 15.93517356758533943976254596031, 16.7219232261966259618105150995, 17.38850031413762598575730562106, 18.765110178588335528694994793762, 19.22627786439771417527640679125, 20.13516815991213297833273440226, 20.91165212771226810154862594702, 21.313276226843791511417691876793, 22.414362713418213594503476498876

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