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Properties

Label 1-967-967.18-r0-0-0
Degree $1$
Conductor $967$
Sign $0.778 + 0.627i$
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  + (−0.395 + 0.918i)2-s + (0.951 + 0.307i)3-s + (−0.687 − 0.726i)4-s + (0.719 − 0.694i)5-s + (−0.658 + 0.752i)6-s + (0.401 − 0.915i)7-s + (0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (0.353 + 0.935i)10-s + (−0.984 − 0.174i)11-s + (−0.430 − 0.902i)12-s + (0.643 + 0.765i)13-s + (0.682 + 0.730i)14-s + (0.898 − 0.439i)15-s + (−0.0552 + 0.998i)16-s + (0.494 + 0.869i)17-s + ⋯
L(s)  = 1  + (−0.395 + 0.918i)2-s + (0.951 + 0.307i)3-s + (−0.687 − 0.726i)4-s + (0.719 − 0.694i)5-s + (−0.658 + 0.752i)6-s + (0.401 − 0.915i)7-s + (0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (0.353 + 0.935i)10-s + (−0.984 − 0.174i)11-s + (−0.430 − 0.902i)12-s + (0.643 + 0.765i)13-s + (0.682 + 0.730i)14-s + (0.898 − 0.439i)15-s + (−0.0552 + 0.998i)16-s + (0.494 + 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.913175635 + 0.6747806830i\)
\(L(\frac12)\) \(\approx\) \(1.913175635 + 0.6747806830i\)
\(L(1)\) \(\approx\) \(1.321776133 + 0.4155764683i\)
\(L(1)\) \(\approx\) \(1.321776133 + 0.4155764683i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.395 + 0.918i)T \)
3 \( 1 + (0.951 + 0.307i)T \)
5 \( 1 + (0.719 - 0.694i)T \)
7 \( 1 + (0.401 - 0.915i)T \)
11 \( 1 + (-0.984 - 0.174i)T \)
13 \( 1 + (0.643 + 0.765i)T \)
17 \( 1 + (0.494 + 0.869i)T \)
19 \( 1 + (0.818 + 0.574i)T \)
23 \( 1 + (0.425 - 0.905i)T \)
29 \( 1 + (-0.750 - 0.660i)T \)
31 \( 1 + (0.413 + 0.910i)T \)
37 \( 1 + (-0.889 + 0.457i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (-0.628 - 0.777i)T \)
53 \( 1 + (-0.158 - 0.987i)T \)
59 \( 1 + (-0.927 - 0.374i)T \)
61 \( 1 + (0.538 - 0.842i)T \)
67 \( 1 + (-0.995 + 0.0974i)T \)
71 \( 1 + (0.993 + 0.116i)T \)
73 \( 1 + (-0.158 + 0.987i)T \)
79 \( 1 + (-0.870 + 0.491i)T \)
83 \( 1 + (0.919 - 0.392i)T \)
89 \( 1 + (0.581 + 0.813i)T \)
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

−21.226846038695380948486493361639, −20.93887164692167742025456686797, −20.24598124774271920814588433614, −19.09372551909323007393374235574, −18.62342618531671352573905663181, −17.981378325136632073734994350082, −17.5761738514484570217281103805, −15.942950942994476610177056770936, −15.26134346290176377283234000563, −14.26712532201210907579748480481, −13.55333661194419343260353379234, −12.98266974950652904304292780595, −12.060312941886143882276314962407, −11.07783864795369472924550289773, −10.32888015789691115400293264293, −9.32844086254893809157772637187, −9.01473849747100462603704112033, −7.70057616135138313404715045337, −7.44411333096532929381604453465, −5.79354035963258420847197541353, −4.98235313233371853373376048864, −3.43258836429586591792015063338, −2.845077879418283630772945120564, −2.205267253009329600755539826832, −1.18939834868862426540570467580, 1.12925981284518428718359639176, 1.91053144691557281588574619595, 3.501506883795811996491617376423, 4.48368172370882791319082471181, 5.16450902767042570912111687941, 6.22502150062368158811211739830, 7.26887668734632573694482997151, 8.191779498468982851311874662247, 8.52232053039289177389583199580, 9.627913093655459391611671802823, 10.15541609664234708865984368958, 10.94142976112955656778719632804, 12.719914110033412037281901657085, 13.40670320149626225309941780203, 14.03725109807826315367883632539, 14.553636382669033420578493543027, 15.65542894106596776495737373914, 16.365806908107521362354833892658, 16.862794616436551865148994901001, 17.850860000795892698624594021517, 18.625810043351575060849555058856, 19.36964919993687183080219773619, 20.4126854682756747581965399364, 20.91188968738918960569457711307, 21.57589941347065682811545253180

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