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Properties

Label 1-967-967.49-r0-0-0
Degree $1$
Conductor $967$
Sign $0.394 - 0.918i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.728 − 0.684i)2-s + (0.710 − 0.703i)3-s + (0.0617 − 0.998i)4-s + (0.643 + 0.765i)5-s + (0.0357 − 0.999i)6-s + (−0.272 + 0.962i)7-s + (−0.638 − 0.769i)8-s + (0.00975 − 0.999i)9-s + (0.993 + 0.116i)10-s + (0.425 + 0.905i)11-s + (−0.658 − 0.752i)12-s + (−0.996 + 0.0844i)13-s + (0.460 + 0.887i)14-s + (0.995 + 0.0909i)15-s + (−0.992 − 0.123i)16-s + (0.874 + 0.485i)17-s + ⋯
L(s)  = 1  + (0.728 − 0.684i)2-s + (0.710 − 0.703i)3-s + (0.0617 − 0.998i)4-s + (0.643 + 0.765i)5-s + (0.0357 − 0.999i)6-s + (−0.272 + 0.962i)7-s + (−0.638 − 0.769i)8-s + (0.00975 − 0.999i)9-s + (0.993 + 0.116i)10-s + (0.425 + 0.905i)11-s + (−0.658 − 0.752i)12-s + (−0.996 + 0.0844i)13-s + (0.460 + 0.887i)14-s + (0.995 + 0.0909i)15-s + (−0.992 − 0.123i)16-s + (0.874 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.394 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.661133775 - 1.753507570i\)
\(L(\frac12)\) \(\approx\) \(2.661133775 - 1.753507570i\)
\(L(1)\) \(\approx\) \(1.887122846 - 0.9118895247i\)
\(L(1)\) \(\approx\) \(1.887122846 - 0.9118895247i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.728 - 0.684i)T \)
3 \( 1 + (0.710 - 0.703i)T \)
5 \( 1 + (0.643 + 0.765i)T \)
7 \( 1 + (-0.272 + 0.962i)T \)
11 \( 1 + (0.425 + 0.905i)T \)
13 \( 1 + (-0.996 + 0.0844i)T \)
17 \( 1 + (0.874 + 0.485i)T \)
19 \( 1 + (0.886 - 0.462i)T \)
23 \( 1 + (0.951 - 0.307i)T \)
29 \( 1 + (0.972 - 0.232i)T \)
31 \( 1 + (0.719 - 0.694i)T \)
37 \( 1 + (0.139 - 0.990i)T \)
41 \( 1 + (-0.917 - 0.398i)T \)
43 \( 1 + (0.771 + 0.636i)T \)
47 \( 1 + (-0.132 - 0.991i)T \)
53 \( 1 + (0.613 - 0.789i)T \)
59 \( 1 + (-0.906 + 0.422i)T \)
61 \( 1 + (0.113 + 0.993i)T \)
67 \( 1 + (0.241 - 0.970i)T \)
71 \( 1 + (-0.957 + 0.288i)T \)
73 \( 1 + (0.613 + 0.789i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.465 + 0.884i)T \)
89 \( 1 + (-0.961 + 0.276i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.84164849378821933114105453506, −21.2014847416006244880273276367, −20.51109411388678511760818456052, −19.90247157371654526173903591524, −18.90522430295042909293346864071, −17.44221143932431985975428021706, −16.883343451268963583288167053846, −16.34211935722136200843964619127, −15.66702313872375068164991133066, −14.49458443314579866635485132513, −13.96255183876185035978393084295, −13.55579827346081301717719083169, −12.5590857124317971775293932144, −11.660315757338808350219921552813, −10.373407484632602590526311328183, −9.63445093720259312515617765729, −8.825676842679744672443824833775, −7.95528775574598450472352485606, −7.18357309599369569188904078018, −6.04717524353264237018317333663, −5.03695621839732192496889761492, −4.59096291245663922898201208768, −3.374961932066091371465946057684, −2.890893491313818167184307614086, −1.22448149019666278347167254424, 1.24476735295872116950914899744, 2.35126072569767525449708343273, 2.655062464773507497251739355365, 3.63586136554199746493819200599, 4.95451936371524688711095735539, 5.880289065417668579419193266188, 6.71757643770495286036143365322, 7.38839734667213661010942153470, 8.834965165354647526981923549920, 9.68270725803455859043075269142, 10.06357569974392659017531907948, 11.47907718480547952712545262865, 12.18397086138099988437444987209, 12.73913305509661389950616890381, 13.63652668739905589779868387850, 14.424816535837262589986396977593, 14.9103758962636421166233409070, 15.50855819458506059893520712865, 17.14915959545168309687324759741, 18.05660237832617049818521137626, 18.62193466392305536066209150657, 19.428115849541451491566881566271, 19.8371170286186419642609382643, 21.09091992942671333735990234225, 21.38691735868843497360529061320

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