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Properties

Label 1-967-967.103-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.459 + 0.888i$
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  + (−0.120 + 0.992i)2-s + (−0.145 − 0.989i)3-s + (−0.971 − 0.238i)4-s + (0.471 + 0.881i)5-s + (0.999 − 0.0260i)6-s + (−0.996 − 0.0844i)7-s + (0.353 − 0.935i)8-s + (−0.957 + 0.288i)9-s + (−0.932 + 0.362i)10-s + (−0.822 − 0.568i)11-s + (−0.0941 + 0.995i)12-s + (−0.0812 − 0.996i)13-s + (0.203 − 0.979i)14-s + (0.803 − 0.595i)15-s + (0.886 + 0.462i)16-s + (−0.883 − 0.468i)17-s + ⋯
L(s)  = 1  + (−0.120 + 0.992i)2-s + (−0.145 − 0.989i)3-s + (−0.971 − 0.238i)4-s + (0.471 + 0.881i)5-s + (0.999 − 0.0260i)6-s + (−0.996 − 0.0844i)7-s + (0.353 − 0.935i)8-s + (−0.957 + 0.288i)9-s + (−0.932 + 0.362i)10-s + (−0.822 − 0.568i)11-s + (−0.0941 + 0.995i)12-s + (−0.0812 − 0.996i)13-s + (0.203 − 0.979i)14-s + (0.803 − 0.595i)15-s + (0.886 + 0.462i)16-s + (−0.883 − 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.459 + 0.888i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2973957041 + 0.4885499039i\)
\(L(\frac12)\) \(\approx\) \(0.2973957041 + 0.4885499039i\)
\(L(1)\) \(\approx\) \(0.6552720972 + 0.2021785597i\)
\(L(1)\) \(\approx\) \(0.6552720972 + 0.2021785597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.120 + 0.992i)T \)
3 \( 1 + (-0.145 - 0.989i)T \)
5 \( 1 + (0.471 + 0.881i)T \)
7 \( 1 + (-0.996 - 0.0844i)T \)
11 \( 1 + (-0.822 - 0.568i)T \)
13 \( 1 + (-0.0812 - 0.996i)T \)
17 \( 1 + (-0.883 - 0.468i)T \)
19 \( 1 + (0.975 - 0.219i)T \)
23 \( 1 + (-0.998 + 0.0585i)T \)
29 \( 1 + (0.737 + 0.675i)T \)
31 \( 1 + (0.999 - 0.0130i)T \)
37 \( 1 + (0.505 + 0.862i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.705 + 0.708i)T \)
47 \( 1 + (-0.982 + 0.187i)T \)
53 \( 1 + (0.995 + 0.0909i)T \)
59 \( 1 + (-0.857 + 0.514i)T \)
61 \( 1 + (-0.715 + 0.699i)T \)
67 \( 1 + (-0.511 + 0.859i)T \)
71 \( 1 + (-0.799 + 0.600i)T \)
73 \( 1 + (0.995 - 0.0909i)T \)
79 \( 1 + (-0.310 - 0.950i)T \)
83 \( 1 + (0.602 - 0.797i)T \)
89 \( 1 + (-0.488 - 0.872i)T \)
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

−21.36825424088216224794652707801, −20.838618688913180799452355298039, −19.92187600744669443286678277466, −19.59902197539062736235169896197, −18.28653255847847990505196818602, −17.62321989916571796672180328887, −16.74097751305603792893094980259, −16.11059630464511822451636310679, −15.33929946199628404085072615363, −13.9685130792784338114452335156, −13.51948669762470179096166707096, −12.43995010233846663011126617740, −11.97047796012141492540965051820, −10.87728607701396355074323270863, −9.91465821653080470649026883769, −9.66552371176459366105345864651, −8.87143024818494623716649211211, −7.998128318567983380335335350938, −6.35384433544325539245241366650, −5.431508362477921560657718990685, −4.54082264875152770990360842205, −3.96740063671467289298070215810, −2.77364253105059324743599703298, −1.93761767432385138313447940503, −0.31744730595224250520874688769, 0.96584066052011611895395242632, 2.68276144695608775578740713540, 3.2004752621343451459727416494, 4.90114591256983969641979052633, 5.95570145550593859057732022777, 6.27094766166652964376479372832, 7.21698105240637662124237273876, 7.810498242158154197996577787, 8.77242533409748876075631580288, 9.88246350268588068220458564633, 10.47986087274623077075682005035, 11.64564192051815353967271173377, 12.82875840409134132451965936778, 13.48989593397793572263263076552, 13.81425689272723376847252478513, 14.894597091746604667854981391327, 15.795902364144794513889591278856, 16.39360787941664116224215001338, 17.54915681435259959100192466842, 18.07735155173605213725836410022, 18.47521159621778483825951405651, 19.444451780403777440588008190996, 20.0466970216031959386177830760, 21.62212701189791906705897045716, 22.418472469554959823701170496397

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