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Properties

Label 1-34e2-1156.491-r1-0-0
Degree $1$
Conductor $1156$
Sign $0.978 + 0.205i$
Analytic cond. $124.229$
Root an. cond. $124.229$
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  + (−0.973 + 0.228i)3-s + (−0.873 − 0.486i)5-s + (0.317 − 0.948i)7-s + (0.895 − 0.445i)9-s + (−0.565 − 0.824i)11-s + (0.273 − 0.961i)13-s + (0.961 + 0.273i)15-s + (0.995 − 0.0922i)19-s + (−0.0922 + 0.995i)21-s + (−0.317 + 0.948i)23-s + (0.526 + 0.850i)25-s + (−0.769 + 0.638i)27-s + (−0.824 + 0.565i)29-s + (0.486 + 0.873i)31-s + (0.739 + 0.673i)33-s + ⋯
L(s)  = 1  + (−0.973 + 0.228i)3-s + (−0.873 − 0.486i)5-s + (0.317 − 0.948i)7-s + (0.895 − 0.445i)9-s + (−0.565 − 0.824i)11-s + (0.273 − 0.961i)13-s + (0.961 + 0.273i)15-s + (0.995 − 0.0922i)19-s + (−0.0922 + 0.995i)21-s + (−0.317 + 0.948i)23-s + (0.526 + 0.850i)25-s + (−0.769 + 0.638i)27-s + (−0.824 + 0.565i)29-s + (0.486 + 0.873i)31-s + (0.739 + 0.673i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.978 + 0.205i$
Analytic conductor: \(124.229\)
Root analytic conductor: \(124.229\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1156,\ (1:\ ),\ 0.978 + 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8796721598 + 0.09116837605i\)
\(L(\frac12)\) \(\approx\) \(0.8796721598 + 0.09116837605i\)
\(L(1)\) \(\approx\) \(0.6731618761 - 0.1100953564i\)
\(L(1)\) \(\approx\) \(0.6731618761 - 0.1100953564i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.973 + 0.228i)T \)
5 \( 1 + (-0.873 - 0.486i)T \)
7 \( 1 + (0.317 - 0.948i)T \)
11 \( 1 + (-0.565 - 0.824i)T \)
13 \( 1 + (0.273 - 0.961i)T \)
19 \( 1 + (0.995 - 0.0922i)T \)
23 \( 1 + (-0.317 + 0.948i)T \)
29 \( 1 + (-0.824 + 0.565i)T \)
31 \( 1 + (0.486 + 0.873i)T \)
37 \( 1 + (-0.0461 + 0.998i)T \)
41 \( 1 + (-0.228 - 0.973i)T \)
43 \( 1 + (0.361 + 0.932i)T \)
47 \( 1 + (0.445 - 0.895i)T \)
53 \( 1 + (-0.895 + 0.445i)T \)
59 \( 1 + (0.798 - 0.602i)T \)
61 \( 1 + (-0.990 + 0.138i)T \)
67 \( 1 + (-0.0922 - 0.995i)T \)
71 \( 1 + (0.948 + 0.317i)T \)
73 \( 1 + (0.914 - 0.403i)T \)
79 \( 1 + (-0.638 + 0.769i)T \)
83 \( 1 + (0.526 + 0.850i)T \)
89 \( 1 + (0.273 + 0.961i)T \)
97 \( 1 + (0.948 + 0.317i)T \)
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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

−21.12438735222379337282846900580, −20.38962927872359909082104045492, −19.19546222712861579206664541799, −18.5864205758571742681136719813, −18.206975893924436642581923513486, −17.299816289520990948317624794345, −16.258575890877264729131012678334, −15.75782505247095479435913599145, −15.02155260822270847073357664483, −14.15088491523092758492115490014, −12.97160099228738108806477264907, −12.20624923706327369603926060325, −11.65314019161891189715502300523, −11.073210710502125144880548533080, −10.11417981452996358309009869660, −9.19727599835905701303504041413, −7.99525409442539554647168420915, −7.41099495189607876754988881109, −6.50482833693630718997823656036, −5.69369650517195641433604991597, −4.72292930111383681738686436327, −4.04738231711040803209838520158, −2.63160042634453916927096562363, −1.765313777724933332995563975265, −0.34500602031911827311006479522, 0.66757846305779150155499147804, 1.27907925280433273059344316181, 3.294889923428737589263615980138, 3.82453956998455985788613581647, 5.033860428307494591066623673058, 5.36579161394464619894805492916, 6.5983996153776223302899248864, 7.595111982409751410611599590636, 8.05035700139234943097500238541, 9.2635337727912671465440610300, 10.31164644227659945394801917900, 10.93804014610812196828607942098, 11.535732438415682939875827404092, 12.386408504293559331567062262095, 13.21289506730202562225750194115, 13.94237439408097574943261525741, 15.28116217893362585663850715154, 15.78928730250693913389453884992, 16.46318336324366971419488347308, 17.15667055273110118102397140580, 17.944583542084470818468651706656, 18.67424696781404644211496228827, 19.709672005576037770463844347008, 20.39735942378622225572276650591, 21.03652436494960531458512816041

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