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Properties

Label 1-760-760.499-r1-0-0
Degree $1$
Conductor $760$
Sign $0.0389 - 0.999i$
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  + (0.939 + 0.342i)3-s + (−0.5 + 0.866i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)17-s + (−0.766 + 0.642i)21-s + (0.173 − 0.984i)23-s + (0.5 + 0.866i)27-s + (−0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s + (−0.173 − 0.984i)33-s + 37-s − 39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (−0.5 + 0.866i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)17-s + (−0.766 + 0.642i)21-s + (0.173 − 0.984i)23-s + (0.5 + 0.866i)27-s + (−0.766 − 0.642i)29-s + (0.5 − 0.866i)31-s + (−0.173 − 0.984i)33-s + 37-s − 39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0389 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0389 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.0389 - 0.999i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ 0.0389 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8448518149 - 0.8125507430i\)
\(L(\frac12)\) \(\approx\) \(0.8448518149 - 0.8125507430i\)
\(L(1)\) \(\approx\) \(1.124331890 + 0.09720331869i\)
\(L(1)\) \(\approx\) \(1.124331890 + 0.09720331869i\)

Euler product

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

−22.43600352586324879011209698765, −21.51211762634870619267167758091, −20.47950955049724805113703984692, −19.99134472111678459726881960538, −19.4713001586885929375926973749, −18.358725203756673908038085733514, −17.70688053370482112744980288595, −16.76470812099952595610948054964, −15.688207704158264858906926771260, −15.06074054763529512266312569209, −14.17152654200287982979523869838, −13.31277278746914475114115016015, −12.83590281048732973028046085658, −11.84451363051946689568090503436, −10.55859279503964428440216842497, −9.78148759132474854959267393555, −9.16479316874838742887036914308, −7.90339229826255717812045334226, −7.284491335066516816763605213819, −6.67123515891727729239028203536, −5.11994268666764010981723492135, −4.21011428210012241696738589415, −3.15247943241335997377170459918, −2.33946701127199466750738151849, −1.13776072394566437804676516430, 0.22695842251833919422555996860, 2.117095413395978509068750787778, 2.64055387732146904685514022176, 3.71933897465196804039234463316, 4.70586470441179150711248611083, 5.77571099789958555888531108159, 6.780685507996017923192705686126, 7.929136627486816163308100704973, 8.63834892649614600248924565085, 9.380108135600123209088274647998, 10.186238588974056521352390164, 11.15528987872970042321968807399, 12.26223079157281932954391018109, 13.113612883636007440191842961220, 13.77237498136265641439557907084, 14.92751942166345482452527928675, 15.24982546633288729853694110252, 16.24263402496084067246318109496, 16.93501564071788821035672026617, 18.2635393364757066256815264787, 19.02382431109347879013241953706, 19.42072381845357119886930890589, 20.47393164560745500870150164081, 21.18930854666121096916189799086, 22.05976726390959926241099195939

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