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Properties

Label 1-755-755.289-r0-0-0
Degree $1$
Conductor $755$
Sign $-0.887 + 0.460i$
Analytic cond. $3.50620$
Root an. cond. $3.50620$
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  + (−0.669 − 0.743i)2-s + (0.637 + 0.770i)3-s + (−0.104 + 0.994i)4-s + (0.146 − 0.989i)6-s + (−0.387 + 0.921i)7-s + (0.809 − 0.587i)8-s + (−0.187 + 0.982i)9-s + (−0.348 − 0.937i)11-s + (−0.832 + 0.553i)12-s + (0.268 + 0.963i)13-s + (0.944 − 0.328i)14-s + (−0.978 − 0.207i)16-s + (−0.604 + 0.796i)17-s + (0.855 − 0.518i)18-s + (−0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.637 + 0.770i)3-s + (−0.104 + 0.994i)4-s + (0.146 − 0.989i)6-s + (−0.387 + 0.921i)7-s + (0.809 − 0.587i)8-s + (−0.187 + 0.982i)9-s + (−0.348 − 0.937i)11-s + (−0.832 + 0.553i)12-s + (0.268 + 0.963i)13-s + (0.944 − 0.328i)14-s + (−0.978 − 0.207i)16-s + (−0.604 + 0.796i)17-s + (0.855 − 0.518i)18-s + (−0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(3.50620\)
Root analytic conductor: \(3.50620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 755,\ (0:\ ),\ -0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1213477011 + 0.4976831704i\)
\(L(\frac12)\) \(\approx\) \(0.1213477011 + 0.4976831704i\)
\(L(1)\) \(\approx\) \(0.6863897839 + 0.1614343474i\)
\(L(1)\) \(\approx\) \(0.6863897839 + 0.1614343474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
151 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (0.637 + 0.770i)T \)
7 \( 1 + (-0.387 + 0.921i)T \)
11 \( 1 + (-0.348 - 0.937i)T \)
13 \( 1 + (0.268 + 0.963i)T \)
17 \( 1 + (-0.604 + 0.796i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (-0.570 - 0.821i)T \)
37 \( 1 + (0.699 - 0.714i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (-0.387 - 0.921i)T \)
47 \( 1 + (-0.783 - 0.621i)T \)
53 \( 1 + (-0.0627 + 0.998i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.985 + 0.166i)T \)
67 \( 1 + (0.187 + 0.982i)T \)
71 \( 1 + (0.604 + 0.796i)T \)
73 \( 1 + (0.992 - 0.125i)T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (-0.728 - 0.684i)T \)
89 \( 1 + (0.228 - 0.973i)T \)
97 \( 1 + (-0.944 - 0.328i)T \)
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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

−22.58492175180112485362100750270, −20.81775701796961311004811848761, −20.13712155230979303539990922854, −19.72695010854536896178808619477, −18.71128150065135656531721124918, −17.94476519891750966347338647124, −17.49070078401671180965351106131, −16.42514302039329602446216940510, −15.56016796638564409320056490863, −14.79423422250258736302371848298, −13.965674089207723677333026127427, −13.23307588495447825839642938627, −12.45883248833654718017464254524, −11.093556480659225651149019667550, −10.060383312668271879321748873431, −9.58023641690744469157029047837, −8.18096641372169225494898822813, −7.97457255455915003411740750882, −6.80273066709166229329462689083, −6.45304151689058639515293549473, −5.06497983299602991652796742142, −3.92929851767966924278528332560, −2.57356479135374594758817990370, −1.51681359641402161435349616442, −0.26218237170317762822431211275, 1.86485475773929427475068554526, 2.57874177143092319019170696226, 3.580028396118283976631419256685, 4.32382440335490151215408520018, 5.6346358299063931134526761854, 6.817005628887308312397947104661, 8.225490961874298393685598864460, 8.678794606344284068578421313585, 9.32355000902395112007197076993, 10.2535415918976108436885176041, 11.05990369488398915884316580663, 11.761736673779269568902462291518, 12.96583499835638282566467283646, 13.55181740617772872298275451560, 14.67740507114115564273464109472, 15.674671660311318589514220131746, 16.2589116889145135630158500930, 17.02596817635548884639176827909, 18.22800994551694659551228799595, 18.91143150539971056529790369744, 19.5565093919385215302352572097, 20.22317251944112523360079075347, 21.37672575357817814752413913719, 21.68602296654163314685470947795, 22.110499820196194202274319929898

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