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Properties

Label 1-967-967.135-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.210 - 0.977i$
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  + (−0.533 − 0.845i)2-s + (−0.407 − 0.913i)3-s + (−0.430 + 0.902i)4-s + (−0.0812 − 0.996i)5-s + (−0.555 + 0.831i)6-s + (−0.741 + 0.670i)7-s + (0.993 − 0.116i)8-s + (−0.668 + 0.744i)9-s + (−0.799 + 0.600i)10-s + (−0.998 − 0.0585i)11-s + (0.999 + 0.0260i)12-s + (0.448 − 0.893i)13-s + (0.962 + 0.269i)14-s + (−0.877 + 0.480i)15-s + (−0.628 − 0.777i)16-s + (0.938 + 0.344i)17-s + ⋯
L(s)  = 1  + (−0.533 − 0.845i)2-s + (−0.407 − 0.913i)3-s + (−0.430 + 0.902i)4-s + (−0.0812 − 0.996i)5-s + (−0.555 + 0.831i)6-s + (−0.741 + 0.670i)7-s + (0.993 − 0.116i)8-s + (−0.668 + 0.744i)9-s + (−0.799 + 0.600i)10-s + (−0.998 − 0.0585i)11-s + (0.999 + 0.0260i)12-s + (0.448 − 0.893i)13-s + (0.962 + 0.269i)14-s + (−0.877 + 0.480i)15-s + (−0.628 − 0.777i)16-s + (0.938 + 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.210 - 0.977i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.210 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4406417402 - 0.5458665488i\)
\(L(\frac12)\) \(\approx\) \(0.4406417402 - 0.5458665488i\)
\(L(1)\) \(\approx\) \(0.4908308935 - 0.3892540323i\)
\(L(1)\) \(\approx\) \(0.4908308935 - 0.3892540323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.533 - 0.845i)T \)
3 \( 1 + (-0.407 - 0.913i)T \)
5 \( 1 + (-0.0812 - 0.996i)T \)
7 \( 1 + (-0.741 + 0.670i)T \)
11 \( 1 + (-0.998 - 0.0585i)T \)
13 \( 1 + (0.448 - 0.893i)T \)
17 \( 1 + (0.938 + 0.344i)T \)
19 \( 1 + (0.880 - 0.474i)T \)
23 \( 1 + (-0.145 + 0.989i)T \)
29 \( 1 + (0.279 + 0.960i)T \)
31 \( 1 + (0.471 + 0.881i)T \)
37 \( 1 + (-0.0162 + 0.999i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (0.795 - 0.605i)T \)
47 \( 1 + (0.998 - 0.0520i)T \)
53 \( 1 + (0.291 - 0.956i)T \)
59 \( 1 + (-0.677 - 0.735i)T \)
61 \( 1 + (-0.158 - 0.987i)T \)
67 \( 1 + (0.527 + 0.849i)T \)
71 \( 1 + (0.999 + 0.0390i)T \)
73 \( 1 + (0.291 + 0.956i)T \)
79 \( 1 + (0.867 - 0.497i)T \)
83 \( 1 + (0.304 + 0.952i)T \)
89 \( 1 + (-0.999 + 0.0325i)T \)
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

−22.32732358927329135206179193588, −21.17055083433532808538693464773, −20.46289929183311336026729701528, −19.363218726325859409801790377916, −18.61434520992823581965452892113, −18.07432728737662559551625785064, −16.97482617157375917437667627573, −16.427282837879932382724321818972, −15.757895257880097322248524589456, −15.14934301782178689302348148275, −14.04113184567154117607160048768, −13.80037181718213882790100615907, −12.22390932760974328068698513612, −11.10930691014398693291376872911, −10.41942934754177296338369497768, −9.92297669871181619113322563010, −9.18704316131110834705526149383, −7.90451691167819842497831533810, −7.22118847150753296854700303263, −6.230605564816049281138285034614, −5.72315032071434736127840206647, −4.48202622289457827249758070662, −3.698301054892117718162721722986, −2.55743170868494401263912532671, −0.658908833836172793078219573017, 0.709397136298353436307800964162, 1.54116972300622547338169274719, 2.76124867386770736058519308041, 3.4488024055294185373878689455, 5.18856872507921158842916136433, 5.46555728237251542088890890392, 6.88391329804182905609303510175, 8.00866540591076054437233215648, 8.316450690959753055223001849862, 9.39541000478286569324230246376, 10.21997037293934539533033700976, 11.17647447808007357342495825353, 12.17834286627184559523774043145, 12.467730508959169286443491045612, 13.24767768758990577987731231450, 13.77567798712941902046370383984, 15.60509507757149889100889278757, 16.1383231464308165584351743559, 17.044203334190712211933368609404, 17.79916922143291266746168621766, 18.44947731313940627010250257572, 19.125039027352846708932298915101, 19.91913266819244550748965409366, 20.48280271557601938525793808590, 21.49014724439622568455637343970

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