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Properties

Label 1-967-967.16-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.492 + 0.870i$
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  + (0.328 + 0.944i)2-s + (0.981 + 0.193i)3-s + (−0.783 + 0.620i)4-s + (−0.953 − 0.300i)5-s + (0.139 + 0.990i)6-s + (0.0617 − 0.998i)7-s + (−0.844 − 0.536i)8-s + (0.924 + 0.380i)9-s + (−0.0292 − 0.999i)10-s + (0.279 + 0.960i)11-s + (−0.889 + 0.457i)12-s + (−0.971 + 0.238i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.228 − 0.973i)16-s + (0.126 + 0.991i)17-s + ⋯
L(s)  = 1  + (0.328 + 0.944i)2-s + (0.981 + 0.193i)3-s + (−0.783 + 0.620i)4-s + (−0.953 − 0.300i)5-s + (0.139 + 0.990i)6-s + (0.0617 − 0.998i)7-s + (−0.844 − 0.536i)8-s + (0.924 + 0.380i)9-s + (−0.0292 − 0.999i)10-s + (0.279 + 0.960i)11-s + (−0.889 + 0.457i)12-s + (−0.971 + 0.238i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.228 − 0.973i)16-s + (0.126 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9129324062 + 1.564617299i\)
\(L(\frac12)\) \(\approx\) \(0.9129324062 + 1.564617299i\)
\(L(1)\) \(\approx\) \(1.115675897 + 0.7717905963i\)
\(L(1)\) \(\approx\) \(1.115675897 + 0.7717905963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.328 + 0.944i)T \)
3 \( 1 + (0.981 + 0.193i)T \)
5 \( 1 + (-0.953 - 0.300i)T \)
7 \( 1 + (0.0617 - 0.998i)T \)
11 \( 1 + (0.279 + 0.960i)T \)
13 \( 1 + (-0.971 + 0.238i)T \)
17 \( 1 + (0.126 + 0.991i)T \)
19 \( 1 + (0.919 - 0.392i)T \)
23 \( 1 + (0.996 + 0.0779i)T \)
29 \( 1 + (-0.998 - 0.0585i)T \)
31 \( 1 + (0.754 + 0.655i)T \)
37 \( 1 + (0.771 - 0.636i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (-0.767 + 0.641i)T \)
47 \( 1 + (0.581 + 0.813i)T \)
53 \( 1 + (0.291 + 0.956i)T \)
59 \( 1 + (0.152 + 0.988i)T \)
61 \( 1 + (-0.158 + 0.987i)T \)
67 \( 1 + (-0.945 - 0.325i)T \)
71 \( 1 + (0.653 + 0.756i)T \)
73 \( 1 + (0.291 - 0.956i)T \)
79 \( 1 + (-0.997 - 0.0714i)T \)
83 \( 1 + (0.861 + 0.508i)T \)
89 \( 1 + (0.190 + 0.981i)T \)
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

−21.48016020563478919623246361305, −20.463521697613182987340533137798, −20.05150681782591532715478446917, −18.96100028478972403368955165971, −18.87339167522529158748469528266, −18.14387069320723869822126517354, −16.678739583271747972409003526317, −15.51707128635502648940038948338, −14.98486834436912773294427975134, −14.32365487554280612150268331573, −13.47245508691856281680594046644, −12.62216231801488044648538756814, −11.73752952178963292960663844260, −11.43447077651271588018761865706, −10.05880661975177601295966759022, −9.359285702371401247380396505638, −8.55201939983166503674189160000, −7.82851791586193001991694867672, −6.74984487967888255171459367902, −5.4280055829066998219921340356, −4.58695108706651375638662421510, −3.29937269076125163480114113423, −3.08604481593505682089014477331, −2.07234628646028376565132495055, −0.70596449832211097547702793898, 1.23247093966021993637066957424, 2.87845582599370740692766994135, 3.8392118606666755870383794934, 4.405061392128957674373426273660, 5.09841185372184019708309487263, 6.7955878555493381341485042372, 7.42766522325098216640302787830, 7.79080543579455885765613514032, 8.89061555678801293303580611412, 9.545718341908134970595415618175, 10.52829390492753122451304547260, 11.88410939871168373722940274563, 12.6995423117386030772301043347, 13.3813474983266309808965177540, 14.304995963849764905603611706582, 14.9813167384234454461940244952, 15.39203279086274243715210957249, 16.46530360484355152848849225895, 16.97426522612937378729630066442, 17.90380412374408318420205957886, 19.081878313440287891116592003562, 19.695009888754477781273663867240, 20.37882294978440188680423102471, 21.183537611297803736660875145217, 22.17491172184570627435466976627

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