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Properties

Label 1-967-967.133-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.889 - 0.456i$
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  + (0.460 + 0.887i)2-s + (0.682 − 0.730i)3-s + (−0.576 + 0.816i)4-s + (−0.334 − 0.942i)5-s + (0.962 + 0.269i)6-s + (−0.775 − 0.631i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (0.682 − 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.203 + 0.979i)12-s + (−0.0682 + 0.997i)13-s + (0.203 − 0.979i)14-s + (−0.917 − 0.398i)15-s + (−0.334 − 0.942i)16-s + (−0.917 + 0.398i)17-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)2-s + (0.682 − 0.730i)3-s + (−0.576 + 0.816i)4-s + (−0.334 − 0.942i)5-s + (0.962 + 0.269i)6-s + (−0.775 − 0.631i)7-s + (−0.990 − 0.136i)8-s + (−0.0682 − 0.997i)9-s + (0.682 − 0.730i)10-s + (−0.0682 − 0.997i)11-s + (0.203 + 0.979i)12-s + (−0.0682 + 0.997i)13-s + (0.203 − 0.979i)14-s + (−0.917 − 0.398i)15-s + (−0.334 − 0.942i)16-s + (−0.917 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.889 - 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1181668313 - 0.4897493863i\)
\(L(\frac12)\) \(\approx\) \(0.1181668313 - 0.4897493863i\)
\(L(1)\) \(\approx\) \(0.9898397495 - 0.05656136000i\)
\(L(1)\) \(\approx\) \(0.9898397495 - 0.05656136000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.460 + 0.887i)T \)
3 \( 1 + (0.682 - 0.730i)T \)
5 \( 1 + (-0.334 - 0.942i)T \)
7 \( 1 + (-0.775 - 0.631i)T \)
11 \( 1 + (-0.0682 - 0.997i)T \)
13 \( 1 + (-0.0682 + 0.997i)T \)
17 \( 1 + (-0.917 + 0.398i)T \)
19 \( 1 + (0.682 - 0.730i)T \)
23 \( 1 + (-0.576 + 0.816i)T \)
29 \( 1 + (-0.0682 + 0.997i)T \)
31 \( 1 + (-0.990 - 0.136i)T \)
37 \( 1 + (-0.0682 + 0.997i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.917 - 0.398i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (-0.576 + 0.816i)T \)
59 \( 1 + (-0.576 - 0.816i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (-0.990 + 0.136i)T \)
71 \( 1 + (0.460 - 0.887i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (-0.576 - 0.816i)T \)
83 \( 1 + (0.962 + 0.269i)T \)
89 \( 1 + (-0.990 + 0.136i)T \)
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

−22.19170664142701464514310889446, −21.32193609747083340658101759649, −20.37413060460247485208703847553, −19.91978168883495056058656677470, −19.215787581087621208098590043968, −18.34278209337384539789588113551, −17.79033153264302523675027116658, −16.09165994353255419597780472859, −15.54366870409445713722474571591, −14.80722232447896484275269127717, −14.30764314426724777777213249595, −13.224362832314209699637089752015, −12.55357254664280650102781115302, −11.582233189831721082383658075626, −10.71740922064424708798505208543, −9.94884970911424515671728749024, −9.5510429742367558861081081940, −8.44288652274441336309736728225, −7.41348165981835233432298089487, −6.21411853197879276489648229858, −5.28074599462161577778567360285, −4.22128585981634304133120196137, −3.48553010695442723114652696863, −2.65804429951989369065774087147, −2.11886392693615271968898649853, 0.15869527252520076324776525169, 1.546062664035943307581554355177, 3.11846471170675634084525942311, 3.75792476423486514549104680566, 4.68163602334665879114710775674, 5.86200312241690069045937384902, 6.69106570238178458515159347840, 7.38511023927447847838477762981, 8.24250070864615354508578994883, 9.02280202385510760483585326973, 9.49179438646152983679656770122, 11.28287030678963841785916617560, 12.11009561845283736473146918229, 13.03099453798264722157431985404, 13.46627202670297915109449328432, 14.00853469594269627369626334996, 15.08714751426939974061735043181, 16.0023368241700941970794624264, 16.45724552732729723749459192000, 17.31300221956294185049440879833, 18.1906592445586321638685751079, 19.16920969765829199813735723678, 19.82484508778020925217670125963, 20.512190846585406041506982108147, 21.5848348700957410079532532153

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