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Properties

Label 1-967-967.161-r0-0-0
Degree $1$
Conductor $967$
Sign $0.998 - 0.0618i$
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  + (−0.936 + 0.350i)2-s + (−0.822 − 0.568i)3-s + (0.754 − 0.655i)4-s + (0.672 − 0.739i)5-s + (0.969 + 0.244i)6-s + (0.719 + 0.694i)7-s + (−0.477 + 0.878i)8-s + (0.353 + 0.935i)9-s + (−0.371 + 0.928i)10-s + (−0.511 + 0.859i)11-s + (−0.993 + 0.110i)12-s + (0.999 − 0.0130i)13-s + (−0.917 − 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.139 − 0.990i)16-s + (0.996 + 0.0779i)17-s + ⋯
L(s)  = 1  + (−0.936 + 0.350i)2-s + (−0.822 − 0.568i)3-s + (0.754 − 0.655i)4-s + (0.672 − 0.739i)5-s + (0.969 + 0.244i)6-s + (0.719 + 0.694i)7-s + (−0.477 + 0.878i)8-s + (0.353 + 0.935i)9-s + (−0.371 + 0.928i)10-s + (−0.511 + 0.859i)11-s + (−0.993 + 0.110i)12-s + (0.999 − 0.0130i)13-s + (−0.917 − 0.398i)14-s + (−0.974 + 0.225i)15-s + (0.139 − 0.990i)16-s + (0.996 + 0.0779i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.998 - 0.0618i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.998 - 0.0618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9766396099 + 0.03021989345i\)
\(L(\frac12)\) \(\approx\) \(0.9766396099 + 0.03021989345i\)
\(L(1)\) \(\approx\) \(0.7444752623 + 0.02643874580i\)
\(L(1)\) \(\approx\) \(0.7444752623 + 0.02643874580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.936 + 0.350i)T \)
3 \( 1 + (-0.822 - 0.568i)T \)
5 \( 1 + (0.672 - 0.739i)T \)
7 \( 1 + (0.719 + 0.694i)T \)
11 \( 1 + (-0.511 + 0.859i)T \)
13 \( 1 + (0.999 - 0.0130i)T \)
17 \( 1 + (0.996 + 0.0779i)T \)
19 \( 1 + (0.505 - 0.862i)T \)
23 \( 1 + (0.527 - 0.849i)T \)
29 \( 1 + (-0.724 + 0.689i)T \)
31 \( 1 + (-0.992 - 0.123i)T \)
37 \( 1 + (-0.895 + 0.445i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (0.929 - 0.368i)T \)
47 \( 1 + (0.975 + 0.219i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (0.914 + 0.404i)T \)
61 \( 1 + (-0.877 + 0.480i)T \)
67 \( 1 + (0.389 - 0.921i)T \)
71 \( 1 + (0.165 + 0.986i)T \)
73 \( 1 + (-0.648 - 0.761i)T \)
79 \( 1 + (-0.597 + 0.801i)T \)
83 \( 1 + (0.795 - 0.605i)T \)
89 \( 1 + (0.602 + 0.797i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−21.389655417663376537567348925190, −21.02349589635418591127922357931, −20.524118801094243377674524257719, −19.01703042836888036169427850769, −18.554734379911133340578053220236, −17.754120280433970463229006689160, −17.228640270311657908385768252885, −16.39790136069432619720908403306, −15.80600474783535268226491459227, −14.70031970633648180099362444719, −13.82834531760720153914053469511, −12.8158089817734781055769158072, −11.62792557524111328710366228603, −11.017089133799139244089141732349, −10.59712299034742002271809447215, −9.82634575732517643632340955568, −8.98855481971054420017956977999, −7.79780978177539770741197954383, −7.14557994627257619971343700764, −5.931706138064847248210035961614, −5.50284599035014857557891874320, −3.789284843993496647143510917, −3.31207200380647999940853854439, −1.77976149816798144041614853813, −0.8793168137405094711001349383, 0.97358061051806961634206968327, 1.673252946227874494839981057760, 2.566114696245051236996753859300, 4.71692794691585390998014136272, 5.447709335008727922737713217431, 5.94349606265658823023510525105, 7.07862043250064244809888235108, 7.81610089530430815703876172828, 8.73255517666386487132056135014, 9.417157005929789104351607538276, 10.535346029418221683454925853829, 11.08591470357090328530741641763, 12.1455265636888864069492790108, 12.69234824695540987745069595085, 13.76899086159170418677694523201, 14.7613167949666286523016076731, 15.75343058554824114897798444332, 16.39784671014585700860175543666, 17.22028471023276489077695682596, 17.82024058968335718455475254567, 18.37366094487234243498992499731, 18.944318192627081574640431598725, 20.27492222412381386200865363762, 20.76644823733028444934990954609, 21.59165888119056236537891266434

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