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Properties

Label 1-967-967.406-r0-0-0
Degree $1$
Conductor $967$
Sign $0.956 + 0.291i$
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  + (−0.668 + 0.744i)2-s + (−0.371 − 0.928i)3-s + (−0.107 − 0.994i)4-s + (0.425 + 0.905i)5-s + (0.938 + 0.344i)6-s + (0.909 + 0.416i)7-s + (0.811 + 0.584i)8-s + (−0.724 + 0.689i)9-s + (−0.957 − 0.288i)10-s + (0.951 − 0.307i)11-s + (−0.883 + 0.468i)12-s + (0.951 + 0.307i)13-s + (−0.917 + 0.398i)14-s + (0.682 − 0.730i)15-s + (−0.977 + 0.212i)16-s + (−0.297 − 0.954i)17-s + ⋯
L(s)  = 1  + (−0.668 + 0.744i)2-s + (−0.371 − 0.928i)3-s + (−0.107 − 0.994i)4-s + (0.425 + 0.905i)5-s + (0.938 + 0.344i)6-s + (0.909 + 0.416i)7-s + (0.811 + 0.584i)8-s + (−0.724 + 0.689i)9-s + (−0.957 − 0.288i)10-s + (0.951 − 0.307i)11-s + (−0.883 + 0.468i)12-s + (0.951 + 0.307i)13-s + (−0.917 + 0.398i)14-s + (0.682 − 0.730i)15-s + (−0.977 + 0.212i)16-s + (−0.297 − 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.956 + 0.291i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.956 + 0.291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.250424505 + 0.1859956812i\)
\(L(\frac12)\) \(\approx\) \(1.250424505 + 0.1859956812i\)
\(L(1)\) \(\approx\) \(0.9103726524 + 0.1321089259i\)
\(L(1)\) \(\approx\) \(0.9103726524 + 0.1321089259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.668 + 0.744i)T \)
3 \( 1 + (-0.371 - 0.928i)T \)
5 \( 1 + (0.425 + 0.905i)T \)
7 \( 1 + (0.909 + 0.416i)T \)
11 \( 1 + (0.951 - 0.307i)T \)
13 \( 1 + (0.951 + 0.307i)T \)
17 \( 1 + (-0.297 - 0.954i)T \)
19 \( 1 + (0.987 - 0.155i)T \)
23 \( 1 + (0.710 - 0.703i)T \)
29 \( 1 + (0.833 - 0.552i)T \)
31 \( 1 + (-0.984 - 0.174i)T \)
37 \( 1 + (0.0876 - 0.996i)T \)
41 \( 1 + (0.854 - 0.519i)T \)
43 \( 1 + (-0.932 + 0.362i)T \)
47 \( 1 + (0.560 + 0.828i)T \)
53 \( 1 + (-0.334 + 0.942i)T \)
59 \( 1 + (-0.844 + 0.536i)T \)
61 \( 1 + (0.854 - 0.519i)T \)
67 \( 1 + (-0.984 + 0.174i)T \)
71 \( 1 + (-0.668 - 0.744i)T \)
73 \( 1 + (-0.334 - 0.942i)T \)
79 \( 1 + (-0.945 - 0.325i)T \)
83 \( 1 + (-0.995 + 0.0974i)T \)
89 \( 1 + (-0.984 + 0.174i)T \)
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

−21.50865358948958315182193746868, −20.883352899464206763992256397233, −20.14429685142470789680562443770, −19.84790954048675575490447139855, −18.33602472735406401307068914640, −17.59781613386680514889777084028, −17.15751484038055058846930833393, −16.47269254227301211187478161321, −15.661936523797181840753440885530, −14.55732640236003638880951254150, −13.62450519678213347927921229383, −12.73050587575334198394495252941, −11.73508822595792868478974666398, −11.25626259501819449554254655120, −10.34180702120850804557761283278, −9.66869735541239284526344507656, −8.74382632786540582475201219910, −8.38465362902589229167399238329, −7.04640376322011470573471369287, −5.778892850525019258786350889310, −4.81585252960752329720183730137, −4.07734623141153156331729001348, −3.28171162879432542170845201467, −1.60556622787833588882866764722, −1.08685891191222035098763909670, 0.98130699097987953828450580869, 1.81000130282042647876865907469, 2.845119053313937167501406210855, 4.56851335306316908242165387045, 5.68950145785696432301779529926, 6.18879756328235237505342301049, 7.079707060828586717457801856913, 7.62911013565837814972847083157, 8.75910865371083808358057916079, 9.2855459204723303319139287027, 10.700953175154101555258997166006, 11.22457170764717212058110992060, 11.85597686371281481800528072278, 13.332216916700073342384086750476, 14.19803812077261862792701091768, 14.33691080331542346338514748079, 15.54098228530349776750425620234, 16.430178451597336362220778593925, 17.31411647086878099963626987880, 17.98410468588903410245702554173, 18.389030304402940793029042137914, 19.01335966536044422530966651207, 19.90722088370259587490424854911, 20.93142137399555136895736182839, 22.1244865766153942968569785439

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