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Properties

Label 1-967-967.24-r1-0-0
Degree $1$
Conductor $967$
Sign $0.841 + 0.539i$
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  + (0.560 + 0.828i)2-s + (−0.874 + 0.485i)3-s + (−0.371 + 0.928i)4-s + (−0.833 + 0.552i)5-s + (−0.892 − 0.451i)6-s + (−0.0487 + 0.998i)7-s + (−0.977 + 0.212i)8-s + (0.527 − 0.849i)9-s + (−0.924 − 0.380i)10-s + (−0.107 + 0.994i)11-s + (−0.126 − 0.991i)12-s + (0.107 + 0.994i)13-s + (−0.854 + 0.519i)14-s + (0.460 − 0.887i)15-s + (−0.724 − 0.689i)16-s + (−0.799 − 0.600i)17-s + ⋯
L(s)  = 1  + (0.560 + 0.828i)2-s + (−0.874 + 0.485i)3-s + (−0.371 + 0.928i)4-s + (−0.833 + 0.552i)5-s + (−0.892 − 0.451i)6-s + (−0.0487 + 0.998i)7-s + (−0.977 + 0.212i)8-s + (0.527 − 0.849i)9-s + (−0.924 − 0.380i)10-s + (−0.107 + 0.994i)11-s + (−0.126 − 0.991i)12-s + (0.107 + 0.994i)13-s + (−0.854 + 0.519i)14-s + (0.460 − 0.887i)15-s + (−0.724 − 0.689i)16-s + (−0.799 − 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.841 + 0.539i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.841 + 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1157735636 + 0.03392433970i\)
\(L(\frac12)\) \(\approx\) \(0.1157735636 + 0.03392433970i\)
\(L(1)\) \(\approx\) \(0.3937924360 + 0.6008911242i\)
\(L(1)\) \(\approx\) \(0.3937924360 + 0.6008911242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.560 + 0.828i)T \)
3 \( 1 + (-0.874 + 0.485i)T \)
5 \( 1 + (-0.833 + 0.552i)T \)
7 \( 1 + (-0.0487 + 0.998i)T \)
11 \( 1 + (-0.107 + 0.994i)T \)
13 \( 1 + (0.107 + 0.994i)T \)
17 \( 1 + (-0.799 - 0.600i)T \)
19 \( 1 + (0.668 + 0.744i)T \)
23 \( 1 + (-0.494 - 0.869i)T \)
29 \( 1 + (-0.710 + 0.703i)T \)
31 \( 1 + (0.972 + 0.232i)T \)
37 \( 1 + (-0.993 - 0.116i)T \)
41 \( 1 + (-0.203 - 0.979i)T \)
43 \( 1 + (0.0292 - 0.999i)T \)
47 \( 1 + (0.967 + 0.250i)T \)
53 \( 1 + (-0.0682 - 0.997i)T \)
59 \( 1 + (-0.957 - 0.288i)T \)
61 \( 1 + (0.203 + 0.979i)T \)
67 \( 1 + (-0.972 + 0.232i)T \)
71 \( 1 + (0.560 - 0.828i)T \)
73 \( 1 + (-0.0682 + 0.997i)T \)
79 \( 1 + (0.822 - 0.568i)T \)
83 \( 1 + (-0.608 - 0.793i)T \)
89 \( 1 + (-0.972 + 0.232i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.57840975486670328662971767923, −20.6221134120355023950653595375, −19.78306336691605872855016460279, −19.45159148326269889611222524923, −18.47840226108403949700010021434, −17.57790973578811155968970061727, −16.88913482622019649999576340729, −15.79720417352916940985877864614, −15.33664697499618286201102696680, −13.74256709326774789734007730936, −13.42022905127525531580583863469, −12.67333215723674317832500764081, −11.76969603530338973240606126028, −11.13355226525960659124311474718, −10.63859932189460960437796135844, −9.586305011472142704926185766545, −8.30730270254046176736670688977, −7.54716919213566352020800115740, −6.405961205687965743969055788124, −5.56543819201942524516582011884, −4.696241396478222192565112205574, −3.91819024989167357725912541888, −2.97663804356148393737267532167, −1.42604529386465214358951822611, −0.691066498382451580545456741903, 0.03681876029031899443558571272, 2.17550682546636326473247606575, 3.41184149972676247118096194931, 4.29382009366154113477175626023, 4.94707204928391583202646204896, 5.89605656137742974988600462626, 6.79783360433330995020628435944, 7.273931140555534618974754213066, 8.55723220782188053418711756293, 9.30228982184618811931715228137, 10.39422034583036645822699072547, 11.52091369566997833024460968689, 12.084297200513060624747638095539, 12.50132307505445000094478203451, 13.93230418932922168175212761513, 14.72797270892767819298348050460, 15.49778175149494961447442558597, 15.8727204004979028657400856717, 16.60822104720718173641286489946, 17.65454958620351547270247734980, 18.305008008857508528949264246865, 18.88798498503875960617433171830, 20.3944407474535285394323944387, 21.03938659183292042806116786074, 22.13614840489248375681859858746

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