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Properties

Label 1-967-967.54-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.617 - 0.786i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.898 + 0.439i)2-s + (0.334 + 0.942i)3-s + (0.613 + 0.789i)4-s + (0.715 + 0.699i)5-s + (−0.113 + 0.993i)6-s + (−0.0227 + 0.999i)7-s + (0.203 + 0.979i)8-s + (−0.775 + 0.631i)9-s + (0.334 + 0.942i)10-s + (−0.775 + 0.631i)11-s + (−0.538 + 0.842i)12-s + (0.158 − 0.987i)13-s + (−0.460 + 0.887i)14-s + (−0.419 + 0.907i)15-s + (−0.247 + 0.968i)16-s + (−0.576 + 0.816i)17-s + ⋯
L(s)  = 1  + (0.898 + 0.439i)2-s + (0.334 + 0.942i)3-s + (0.613 + 0.789i)4-s + (0.715 + 0.699i)5-s + (−0.113 + 0.993i)6-s + (−0.0227 + 0.999i)7-s + (0.203 + 0.979i)8-s + (−0.775 + 0.631i)9-s + (0.334 + 0.942i)10-s + (−0.775 + 0.631i)11-s + (−0.538 + 0.842i)12-s + (0.158 − 0.987i)13-s + (−0.460 + 0.887i)14-s + (−0.419 + 0.907i)15-s + (−0.247 + 0.968i)16-s + (−0.576 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.617 - 0.786i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.752027482 + 3.603130341i\)
\(L(\frac12)\) \(\approx\) \(-1.752027482 + 3.603130341i\)
\(L(1)\) \(\approx\) \(1.051340184 + 1.746607975i\)
\(L(1)\) \(\approx\) \(1.051340184 + 1.746607975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.898 + 0.439i)T \)
3 \( 1 + (0.334 + 0.942i)T \)
5 \( 1 + (0.715 + 0.699i)T \)
7 \( 1 + (-0.0227 + 0.999i)T \)
11 \( 1 + (-0.775 + 0.631i)T \)
13 \( 1 + (0.158 - 0.987i)T \)
17 \( 1 + (-0.576 + 0.816i)T \)
19 \( 1 + (-0.983 - 0.181i)T \)
23 \( 1 + (0.990 - 0.136i)T \)
29 \( 1 + (0.775 + 0.631i)T \)
31 \( 1 + (0.746 - 0.665i)T \)
37 \( 1 + (0.158 - 0.987i)T \)
41 \( 1 + (0.917 - 0.398i)T \)
43 \( 1 + (-0.995 + 0.0909i)T \)
47 \( 1 + (0.419 - 0.907i)T \)
53 \( 1 + (0.377 - 0.926i)T \)
59 \( 1 + (0.377 + 0.926i)T \)
61 \( 1 + (0.803 + 0.595i)T \)
67 \( 1 + (-0.203 + 0.979i)T \)
71 \( 1 + (-0.0682 + 0.997i)T \)
73 \( 1 + (0.377 + 0.926i)T \)
79 \( 1 + (-0.377 - 0.926i)T \)
83 \( 1 + (0.113 - 0.993i)T \)
89 \( 1 + (0.949 - 0.313i)T \)
97 \( 1 + T \)
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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

−20.96773800935177352499763386036, −20.46670115564162708887838890061, −19.53927476798013232845898968384, −18.98631378127263478871762882426, −18.02317543587426782813780031083, −17.05146657167194587505869184388, −16.361307793253673545452824540515, −15.31854564065016433317575001756, −14.06418505247637506223097858325, −13.80412529971457831731668596487, −13.19117570674597810717381941810, −12.529119296624523062662706667315, −11.53177006482151523665787248054, −10.78062084078405073474111752714, −9.765147120816869612301785804977, −8.83743706837902590410472810312, −7.84002873684281344955425593434, −6.65519223902035747777292513092, −6.32880335736642433875004601990, −5.06227030983169655827392694225, −4.3614023420959543056687225475, −3.09761247606975785001542351915, −2.27572780513929033072816610341, −1.29289559364592307372400196526, −0.52701198017068343121892538422, 2.30074638346984430364981741392, 2.554411812028497095105918863050, 3.587715755225488751284084380, 4.69341779599913216168124056491, 5.45975221479843827420237491360, 6.0826119615833653079604801483, 7.12558835176940857641764288565, 8.299325712835180678885982092019, 8.881946991238000466268380840475, 10.19952960969948177183332939916, 10.66312703829284211932958912535, 11.6051921585135645499878858975, 12.93715975481359881792977652226, 13.15746409465667980184344582760, 14.476610059973246946323396793993, 14.994865356938160405771238018667, 15.37670633338058911450547506317, 16.188859978216580451383234495770, 17.38096529171685648125813036926, 17.76545408117907512070292298227, 18.99648197338101492038159697900, 19.98491972392185753818775632937, 20.93835919481333593335896549452, 21.41210353664401161900892835517, 21.94555625017084936627731177766

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