Properties

Label 1-967-967.822-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.999 + 0.00912i$
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  + (0.995 + 0.0909i)2-s + (0.0682 + 0.997i)3-s + (0.983 + 0.181i)4-s + (0.158 + 0.987i)5-s + (−0.0227 + 0.999i)6-s + (0.949 + 0.313i)7-s + (0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (0.0682 + 0.997i)10-s + (−0.990 + 0.136i)11-s + (−0.113 + 0.993i)12-s + (−0.613 + 0.789i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.934 + 0.356i)16-s + (0.682 − 0.730i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0909i)2-s + (0.0682 + 0.997i)3-s + (0.983 + 0.181i)4-s + (0.158 + 0.987i)5-s + (−0.0227 + 0.999i)6-s + (0.949 + 0.313i)7-s + (0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (0.0682 + 0.997i)10-s + (−0.990 + 0.136i)11-s + (−0.113 + 0.993i)12-s + (−0.613 + 0.789i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.934 + 0.356i)16-s + (0.682 − 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.999 + 0.00912i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (822, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.999 + 0.00912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01970196843 + 4.316346413i\)
\(L(\frac12)\) \(\approx\) \(0.01970196843 + 4.316346413i\)
\(L(1)\) \(\approx\) \(1.547633396 + 1.455239471i\)
\(L(1)\) \(\approx\) \(1.547633396 + 1.455239471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.995 + 0.0909i)T \)
3 \( 1 + (0.0682 + 0.997i)T \)
5 \( 1 + (0.158 + 0.987i)T \)
7 \( 1 + (0.949 + 0.313i)T \)
11 \( 1 + (-0.990 + 0.136i)T \)
13 \( 1 + (-0.613 + 0.789i)T \)
17 \( 1 + (0.682 - 0.730i)T \)
19 \( 1 + (0.829 - 0.557i)T \)
23 \( 1 + (0.334 + 0.942i)T \)
29 \( 1 + (0.990 + 0.136i)T \)
31 \( 1 + (-0.715 + 0.699i)T \)
37 \( 1 + (-0.613 + 0.789i)T \)
41 \( 1 + (-0.854 - 0.519i)T \)
43 \( 1 + (-0.291 + 0.956i)T \)
47 \( 1 + (0.974 - 0.225i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (-0.648 - 0.761i)T \)
61 \( 1 + (-0.877 + 0.480i)T \)
67 \( 1 + (-0.962 + 0.269i)T \)
71 \( 1 + (-0.576 - 0.816i)T \)
73 \( 1 + (-0.648 - 0.761i)T \)
79 \( 1 + (0.648 + 0.761i)T \)
83 \( 1 + (0.0227 - 0.999i)T \)
89 \( 1 + (0.247 - 0.968i)T \)
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

−20.94377519671583477086148710833, −20.52330497434715241232260471837, −19.90140036606637076560040579997, −18.916740946209810406722469147, −17.991179901824373362406133173, −17.12504438802749175349347807346, −16.51666050005275864226440751111, −15.40135530186930871472063228201, −14.55390511678856865334942028724, −13.843704388982270713943349626535, −13.140653204431612740646399639497, −12.39729530010809620354243090991, −12.00216059781002072941578843953, −10.85605656559924190124713800932, −10.11110183594592188921877340066, −8.53371377568382954212167143364, −7.85908578710330804930437901834, −7.31639355846204908974836497412, −5.922085702442500098926870062373, −5.38134697851668823859618319592, −4.65618947962447883158649801703, −3.374000907833536953034204687150, −2.31878972954265039225689480934, −1.47160336717103718587163887996, −0.56054626959989400956864024516, 1.77563188255420143253483565853, 2.85086408492492013658518087646, 3.2791773410277423953882033587, 4.71984711519246489476575736693, 5.04329595197721791058318112200, 5.94714351278505376637751458419, 7.206776617879215662120236403331, 7.7381275879376623426238660431, 9.06017280052756895832638039751, 10.12211597847026667998098406274, 10.75111871604573561372449605761, 11.59005449603723394341966534700, 12.045558500284702726851442159795, 13.65946372653642460601610028048, 14.05042889287040624350530721211, 14.80418934399123663246331606928, 15.44671504739531969105183251212, 16.03232818329185613504635609034, 17.08644025192051353987399729540, 17.89614283661416718802708713074, 18.87510953698647951920971573131, 19.91289298522692344200674780716, 20.721559076012499324397448126874, 21.45483838478167310191179959202, 21.75091691033544413831982506106

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