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Properties

Label 1-967-967.33-r1-0-0
Degree $1$
Conductor $967$
Sign $0.463 + 0.886i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.932 − 0.362i)2-s + (0.998 + 0.0585i)3-s + (0.737 + 0.675i)4-s + (0.511 + 0.859i)5-s + (−0.909 − 0.416i)6-s + (0.984 + 0.174i)7-s + (−0.442 − 0.896i)8-s + (0.993 + 0.116i)9-s + (−0.165 − 0.986i)10-s + (0.527 + 0.849i)11-s + (0.696 + 0.717i)12-s + (−0.527 + 0.849i)13-s + (−0.854 − 0.519i)14-s + (0.460 + 0.887i)15-s + (0.0876 + 0.996i)16-s + (0.981 − 0.193i)17-s + ⋯
L(s)  = 1  + (−0.932 − 0.362i)2-s + (0.998 + 0.0585i)3-s + (0.737 + 0.675i)4-s + (0.511 + 0.859i)5-s + (−0.909 − 0.416i)6-s + (0.984 + 0.174i)7-s + (−0.442 − 0.896i)8-s + (0.993 + 0.116i)9-s + (−0.165 − 0.986i)10-s + (0.527 + 0.849i)11-s + (0.696 + 0.717i)12-s + (−0.527 + 0.849i)13-s + (−0.854 − 0.519i)14-s + (0.460 + 0.887i)15-s + (0.0876 + 0.996i)16-s + (0.981 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.463 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.582055551 + 1.563399525i\)
\(L(\frac12)\) \(\approx\) \(2.582055551 + 1.563399525i\)
\(L(1)\) \(\approx\) \(1.352981062 + 0.2737077433i\)
\(L(1)\) \(\approx\) \(1.352981062 + 0.2737077433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.932 - 0.362i)T \)
3 \( 1 + (0.998 + 0.0585i)T \)
5 \( 1 + (0.511 + 0.859i)T \)
7 \( 1 + (0.984 + 0.174i)T \)
11 \( 1 + (0.527 + 0.849i)T \)
13 \( 1 + (-0.527 + 0.849i)T \)
17 \( 1 + (0.981 - 0.193i)T \)
19 \( 1 + (-0.874 - 0.485i)T \)
23 \( 1 + (0.822 - 0.568i)T \)
29 \( 1 + (0.945 + 0.325i)T \)
31 \( 1 + (-0.977 - 0.212i)T \)
37 \( 1 + (0.107 - 0.994i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (0.407 - 0.913i)T \)
47 \( 1 + (0.0292 + 0.999i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (0.494 + 0.869i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (0.977 - 0.212i)T \)
71 \( 1 + (-0.932 + 0.362i)T \)
73 \( 1 + (-0.0682 - 0.997i)T \)
79 \( 1 + (0.957 - 0.288i)T \)
83 \( 1 + (0.892 - 0.451i)T \)
89 \( 1 + (0.977 - 0.212i)T \)
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

−21.025290132782434756895669363407, −20.6334355034133678291776043103, −19.69957514634935693641225923053, −19.18667745215466385323388481146, −18.25070981879802983852528497609, −17.39197818285255562392825406204, −16.862476879280215884909087943026, −16.00414678446650366598861563847, −14.95007759118212299019114509363, −14.48871437947609176959392339899, −13.67051615339605505189533416914, −12.65987504847042114191151684756, −11.72145632464428699452167704692, −10.560770223686471304449280053928, −9.908818692453597487659968008266, −9.00522725497273114403755600556, −8.31089140565720136419150167806, −7.93611694646827395050854086441, −6.849264115915683194557269823816, −5.68285424250454526045905559428, −4.931949492349675113522945707988, −3.60255646037539728881606036321, −2.38259547162406408583625946809, −1.44144583929829303464345222754, −0.79781145504943284938837007708, 1.2609063491317103826451540866, 2.11154304607209923511532571156, 2.636196963138247320199218081630, 3.808821797873861732164044996683, 4.80582055189413622944994483027, 6.45049938137044262451101154503, 7.218876086673226411327998669575, 7.7766291607274483422305239243, 8.95807844192398402869520887737, 9.33815559606936207804010199078, 10.311851439608228265230192474415, 10.93709282582431180523783487417, 11.97957687764816194296119272596, 12.74851789727535546183735582737, 13.97911193647338617138034495778, 14.73430814605395980572458266834, 15.02229244538765516467673780263, 16.296844170841722808333727494241, 17.25823854202428064850814017066, 17.85466389133262405724661758988, 18.748206621017702047435777908985, 19.14287242467127712327764329160, 20.085608971512002236148858465137, 20.82447977724177791574837204656, 21.51506125352013272922622267554

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