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Properties

Label 1-760-760.523-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.647 - 0.761i$
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  + (−0.984 − 0.173i)3-s + (0.866 − 0.5i)7-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.342 + 0.939i)17-s + (−0.939 + 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.866 − 0.5i)27-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.642 − 0.766i)33-s − i·37-s + 39-s + (−0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (0.866 − 0.5i)7-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.342 + 0.939i)17-s + (−0.939 + 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.866 − 0.5i)27-s + (0.939 + 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.642 − 0.766i)33-s − i·37-s + 39-s + (−0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.647 - 0.761i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.647 - 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2241036728 - 0.4848945499i\)
\(L(\frac12)\) \(\approx\) \(0.2241036728 - 0.4848945499i\)
\(L(1)\) \(\approx\) \(0.7300886581 - 0.05601734690i\)
\(L(1)\) \(\approx\) \(0.7300886581 - 0.05601734690i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−22.36787169965423255790977564233, −21.70539893185358543704795010575, −21.18511976258059484539207486131, −20.21750331236649157600481077095, −19.0383768260606650820322565677, −18.34412334074639168126856527735, −17.635067483073328823646020011863, −16.932298989924712113462079092352, −15.963777656329836079376657225862, −15.41563875759182636799664889614, −14.325189514960366570379292836, −13.516317008263734716758461407656, −12.2255928338720736625940495362, −11.8787889897060751505950808212, −10.96060805344403562668671318366, −10.19403853157810449863082165846, −9.247266225066499744635940885349, −8.112218152147344683605519692721, −7.311717241346060259707100262030, −6.19562806880067125939912721120, −5.203005020487103109018894045735, −4.90469754265679771727329199206, −3.47537167631601707783792742358, −2.24852243002559790811767181518, −0.9772997041895131307815176315, 0.16186292515882233647105005170, 1.44152865277555904221926496314, 2.33874666334759757269366210077, 4.11666697753907482141927151840, 4.73176180825702905943342093195, 5.58841698525469953694066987105, 6.67796048083866473441271515371, 7.50143200493664000956447237674, 8.183933746074841186425519886033, 9.70526783206552936205887189227, 10.389763025467483104153300651058, 11.07955789537278037473353058271, 12.15131019756000698868241424848, 12.55954787859024527465098057595, 13.66003938243909584425908370483, 14.68463946258807901307138771567, 15.32710992569526359224456473457, 16.605890214960668645792077029774, 16.97379505462419438343379431378, 17.95571380456470881182193186514, 18.31059328994554651012089173633, 19.5622421305614451458108195021, 20.284078916535997101232077474526, 21.38804330204265357678716878691, 21.79297010324585633159547271905

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