Properties

Label 1-967-967.20-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: $\chi_{967} (20, \cdot )$
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

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

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