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Properties

Label 1-967-967.192-r1-0-0
Degree $1$
Conductor $967$
Sign $0.999 + 0.00655i$
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  + (−0.945 − 0.325i)2-s + (−0.353 − 0.935i)3-s + (0.787 + 0.615i)4-s + (−0.909 − 0.416i)5-s + (0.0292 + 0.999i)6-s + (0.883 + 0.468i)7-s + (−0.544 − 0.838i)8-s + (−0.750 + 0.660i)9-s + (0.724 + 0.689i)10-s + (−0.477 − 0.878i)11-s + (0.297 − 0.954i)12-s + (0.477 − 0.878i)13-s + (−0.682 − 0.730i)14-s + (−0.0682 + 0.997i)15-s + (0.241 + 0.970i)16-s + (0.987 + 0.155i)17-s + ⋯
L(s)  = 1  + (−0.945 − 0.325i)2-s + (−0.353 − 0.935i)3-s + (0.787 + 0.615i)4-s + (−0.909 − 0.416i)5-s + (0.0292 + 0.999i)6-s + (0.883 + 0.468i)7-s + (−0.544 − 0.838i)8-s + (−0.750 + 0.660i)9-s + (0.724 + 0.689i)10-s + (−0.477 − 0.878i)11-s + (0.297 − 0.954i)12-s + (0.477 − 0.878i)13-s + (−0.682 − 0.730i)14-s + (−0.0682 + 0.997i)15-s + (0.241 + 0.970i)16-s + (0.987 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.999 + 0.00655i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.999 + 0.00655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7271775924 + 0.002385001404i\)
\(L(\frac12)\) \(\approx\) \(0.7271775924 + 0.002385001404i\)
\(L(1)\) \(\approx\) \(0.5386276250 - 0.2337746439i\)
\(L(1)\) \(\approx\) \(0.5386276250 - 0.2337746439i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.945 - 0.325i)T \)
3 \( 1 + (-0.353 - 0.935i)T \)
5 \( 1 + (-0.909 - 0.416i)T \)
7 \( 1 + (0.883 + 0.468i)T \)
11 \( 1 + (-0.477 - 0.878i)T \)
13 \( 1 + (0.477 - 0.878i)T \)
17 \( 1 + (0.987 + 0.155i)T \)
19 \( 1 + (0.511 - 0.859i)T \)
23 \( 1 + (0.442 + 0.896i)T \)
29 \( 1 + (-0.0487 + 0.998i)T \)
31 \( 1 + (-0.696 + 0.717i)T \)
37 \( 1 + (-0.389 - 0.921i)T \)
41 \( 1 + (-0.460 - 0.887i)T \)
43 \( 1 + (0.957 + 0.288i)T \)
47 \( 1 + (0.822 - 0.568i)T \)
53 \( 1 + (-0.775 + 0.631i)T \)
59 \( 1 + (-0.977 + 0.212i)T \)
61 \( 1 + (0.460 + 0.887i)T \)
67 \( 1 + (0.696 + 0.717i)T \)
71 \( 1 + (-0.945 + 0.325i)T \)
73 \( 1 + (-0.775 - 0.631i)T \)
79 \( 1 + (-0.972 - 0.232i)T \)
83 \( 1 + (-0.967 + 0.250i)T \)
89 \( 1 + (0.696 + 0.717i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.21741885440312070799096971436, −20.531633330087655186382038990649, −20.27755582898959254236489047355, −18.838100769394693907784652414762, −18.54221810271492195598074418683, −17.458256001044988188113788123112, −16.79558666466793124931734619478, −16.113850687170059536719099243620, −15.38243269899945434194418595496, −14.65379379840041086103832734596, −14.1610133903070559954970950014, −12.26801685739764239027082151704, −11.558884197545518154190436800338, −10.96988304658721669468850431558, −10.20936960500032321559252760785, −9.53183045251763332818771829688, −8.38623116069866279650953994761, −7.774447275295371658143362744292, −6.95547026533303430698854726526, −5.89785033022772207658900097994, −4.83406377166281406357914064320, −4.082398977198839958571554266825, −2.92231265241992377281362242090, −1.55172158624671549119222202813, −0.312246686383400967520163920689, 0.83481860396497171168502430090, 1.370868673767835620791502372850, 2.74835059797366177143003176964, 3.5150880824696500657366992583, 5.22323255214160316835659101670, 5.77175527104034202384770961174, 7.39083352013805991573286099419, 7.522664762954022505458575727696, 8.583578364569020063806069387707, 8.89168957367779515589906794057, 10.6620901824885882181329901177, 11.053135981077219451862282552040, 11.846210620394482128976604741561, 12.473157583131902560966214520640, 13.24496598158657528517589800924, 14.42808727516967081421416406148, 15.60070171898519344209402446742, 16.09469450557787199338297376964, 17.0915858180046843232640007231, 17.76811166666187940630478038556, 18.48326561269258592828215511968, 19.040816061113613951488183932478, 19.80970556879308326703051670031, 20.52339740564687750725110514721, 21.357425084743564638220721327662

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