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Properties

Label 1-760-760.397-r1-0-0
Degree $1$
Conductor $760$
Sign $0.944 - 0.328i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.228966893 - 0.2077331509i\)
\(L(\frac12)\) \(\approx\) \(1.228966893 - 0.2077331509i\)
\(L(1)\) \(\approx\) \(0.7960021852 + 0.006424359698i\)
\(L(1)\) \(\approx\) \(0.7960021852 + 0.006424359698i\)

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

−22.391080309598802490146402425757, −21.60140194397427009153786120476, −20.88125218529167681240505624973, −19.56085278960256983446797126711, −18.88295248743743712392566759774, −18.39137164935203883534846869247, −17.149603500467185421180431999424, −16.76282283423146826584908815392, −15.78639357819704253699558941200, −15.23309377240896093504578726625, −13.83795195930454623196784969825, −13.053842267245932094704216978322, −12.41892561843494257916416510815, −11.38537688852990190535108768242, −10.8803420799749708052787711249, −9.78885354784714982850580978014, −8.953774034736542943243950389248, −7.88067617519820791749330014008, −6.74239935461129027249624078371, −5.952531366899991817743046608228, −5.546531146178438764168533543640, −4.03892025021516734673230225834, −3.280180702873376624975243521353, −1.738222090042837535634555197308, −0.67638840068173809229281963827, 0.53608048256307951858359850915, 1.56960517591200467406758515764, 3.1966392213859035250437209397, 4.12477131652690945355981485309, 4.99801849403015848566394977595, 6.0988973990775198905396927674, 6.7993026322564813352100626534, 7.50641907875093044736911455245, 9.10747640019572899632901914048, 9.6647964238840782489496104292, 10.671321988045863839912002672415, 11.276299605395645683777367056793, 12.39109078306734252558789179655, 12.86175415955763179513470696780, 13.91609990191541977189899332764, 14.93998596779963932431143690673, 15.97870708551510299590234021247, 16.436861494627215003605892068172, 17.19220621435542826887274596687, 18.15310898442392778152619751081, 18.710274973197119356511032834088, 19.85462487717105038212496078077, 20.57660123510717353501716030785, 21.45633754883513036467192897617, 22.38837955522785905086178358740

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