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Properties

Label 1-760-760.347-r0-0-0
Degree $1$
Conductor $760$
Sign $0.829 + 0.558i$
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  + (−0.342 + 0.939i)3-s + (0.866 + 0.5i)7-s + (−0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)17-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (0.866 − 0.5i)27-s + (0.766 + 0.642i)29-s + (0.5 − 0.866i)31-s + (0.984 − 0.173i)33-s − i·37-s + 39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (0.866 + 0.5i)7-s + (−0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)17-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (0.866 − 0.5i)27-s + (0.766 + 0.642i)29-s + (0.5 − 0.866i)31-s + (0.984 − 0.173i)33-s − i·37-s + 39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(3.52942\)
Root analytic conductor: \(3.52942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (0:\ ),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289905603 + 0.3937540678i\)
\(L(\frac12)\) \(\approx\) \(1.289905603 + 0.3937540678i\)
\(L(1)\) \(\approx\) \(1.015605782 + 0.2421904685i\)
\(L(1)\) \(\approx\) \(1.015605782 + 0.2421904685i\)

Euler product

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

−22.61281536549681763738976902738, −21.414808700004334433566880424509, −20.71279158267286107436816696955, −19.89127279964072642479893152771, −18.93193887570079774327087689897, −18.34349692368711108219787201494, −17.407174404451786502572875661639, −17.015109925643057246218001510899, −15.93411651697994262899023602739, −14.77722570824573750482849544296, −14.0455229880646553937283620307, −13.384867003531812316670169875456, −12.28535663721184707920386472793, −11.777357739618659211881333264165, −10.866182831884854646716044899985, −9.95869075138189034834049302916, −8.77080803509479286440510223673, −7.765896928147569036523097131525, −7.21752563671038936762361015403, −6.38690054618726679484511374367, −5.03388891170133040179722556078, −4.6223280633438256503889696831, −2.93305422299856142453508808212, −1.93292735798419926767731461961, −0.980132611952826404081949597706, 0.89487778318119672429697151283, 2.54565629074807208943664957027, 3.42341083852151022107866788759, 4.54832931505553352650666774171, 5.48623397781278747788239745676, 5.86271199278118082833538802555, 7.44090556819381895886243798027, 8.4190604251367658690868165494, 9.00539012318955579883051972427, 10.32449315735358664430252672327, 10.68678357550628898462635721458, 11.65903524031717465712611985882, 12.4169403738931580213513955215, 13.55890663835802837178848210585, 14.625476793104250217412925716191, 15.14468940885231639827558481171, 15.90573029643196541562551049249, 16.86740182995173161834362918391, 17.490160111504376982653760876942, 18.34227746563693558722312845461, 19.2764276042414133047125208403, 20.294683932661648596914773177363, 21.15413427965674949367279479851, 21.49692295597800004852338595402, 22.35653721674064865439944221136

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