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Properties

Label 1-760-760.349-r0-0-0
Degree $1$
Conductor $760$
Sign $0.671 + 0.740i$
Analytic cond. $3.52942$
Root an. cond. $3.52942$
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  + (−0.5 + 0.866i)3-s − 7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)21-s + (0.5 + 0.866i)23-s + 27-s + (0.5 + 0.866i)29-s + 31-s + (0.5 − 0.866i)33-s + 37-s + 39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s − 7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)21-s + (0.5 + 0.866i)23-s + 27-s + (0.5 + 0.866i)29-s + 31-s + (0.5 − 0.866i)33-s + 37-s + 39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(3.52942\)
Root analytic conductor: \(3.52942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (0:\ ),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7481823205 + 0.3315832018i\)
\(L(\frac12)\) \(\approx\) \(0.7481823205 + 0.3315832018i\)
\(L(1)\) \(\approx\) \(0.7315685678 + 0.1706194328i\)
\(L(1)\) \(\approx\) \(0.7315685678 + 0.1706194328i\)

Euler product

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

−22.37867515148790673416767235587, −21.67634663949445233721916517848, −20.6792980559298050601757519590, −19.599164804011401161857021924777, −18.95273310010043556888423893641, −18.52841533250510247249049509811, −17.329429518963400420336071883567, −16.77947483706025952326313742414, −15.97117464421499063821076523975, −14.9878861068229905200407495732, −13.83679849790946175423087203348, −13.20906589678933733338593255315, −12.41143282997043013006533949801, −11.82509782424347673548470615237, −10.6105660302382479258058091524, −10.02443338814977457041560270556, −8.77566156490906380808598356712, −7.87370031668340530641109388340, −6.95025096558202671604898251846, −6.26768743702203485714760897900, −5.391275457799997424575076182, −4.28284459306037007077595463783, −2.89036087106966132348854900603, −2.105929797373112216215715133587, −0.6310557651606227028645819625, 0.74687208418656560523789620584, 2.86088600998565221333014124043, 3.22775946001053766523206261187, 4.66110763163738958684739037590, 5.33432929441658494087294286142, 6.20984356196592975801938522695, 7.26567966222243011019982484135, 8.30323832302125018802542731283, 9.56836490966717861825383259302, 9.91717640723523087912645156659, 10.78616186365426932486907944747, 11.72549216665816625005996746564, 12.64270576880746681511453713187, 13.359997817975650904901165731780, 14.548601713981236117193873084237, 15.43632194155380849082504945380, 15.990219384752673358590404079083, 16.669341960671310542084771394472, 17.62941078507165969115311512806, 18.33901695542864325689361111512, 19.41346902668061102512842960393, 20.20916523520933545953194542944, 20.99692687623134816498239565494, 21.76022216425804574497064228095, 22.55912093342316910529292215421

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