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Properties

Label 1-967-967.178-r1-0-0
Degree $1$
Conductor $967$
Sign $0.958 + 0.284i$
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  + (−0.608 + 0.793i)2-s + (−0.938 − 0.344i)3-s + (−0.260 − 0.965i)4-s + (−0.996 + 0.0779i)5-s + (0.844 − 0.536i)6-s + (−0.494 − 0.869i)7-s + (0.924 + 0.380i)8-s + (0.763 + 0.646i)9-s + (0.544 − 0.838i)10-s + (0.981 − 0.193i)11-s + (−0.0876 + 0.996i)12-s + (−0.981 − 0.193i)13-s + (0.990 + 0.136i)14-s + (0.962 + 0.269i)15-s + (−0.864 + 0.502i)16-s + (0.389 − 0.921i)17-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.938 − 0.344i)3-s + (−0.260 − 0.965i)4-s + (−0.996 + 0.0779i)5-s + (0.844 − 0.536i)6-s + (−0.494 − 0.869i)7-s + (0.924 + 0.380i)8-s + (0.763 + 0.646i)9-s + (0.544 − 0.838i)10-s + (0.981 − 0.193i)11-s + (−0.0876 + 0.996i)12-s + (−0.981 − 0.193i)13-s + (0.990 + 0.136i)14-s + (0.962 + 0.269i)15-s + (−0.864 + 0.502i)16-s + (0.389 − 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.958 + 0.284i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.958 + 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7370212668 + 0.1069580256i\)
\(L(\frac12)\) \(\approx\) \(0.7370212668 + 0.1069580256i\)
\(L(1)\) \(\approx\) \(0.5213196333 + 0.04075304370i\)
\(L(1)\) \(\approx\) \(0.5213196333 + 0.04075304370i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.608 + 0.793i)T \)
3 \( 1 + (-0.938 - 0.344i)T \)
5 \( 1 + (-0.996 + 0.0779i)T \)
7 \( 1 + (-0.494 - 0.869i)T \)
11 \( 1 + (0.981 - 0.193i)T \)
13 \( 1 + (-0.981 - 0.193i)T \)
17 \( 1 + (0.389 - 0.921i)T \)
19 \( 1 + (0.995 - 0.0974i)T \)
23 \( 1 + (0.883 - 0.468i)T \)
29 \( 1 + (0.407 - 0.913i)T \)
31 \( 1 + (0.279 + 0.960i)T \)
37 \( 1 + (0.799 + 0.600i)T \)
41 \( 1 + (0.334 + 0.942i)T \)
43 \( 1 + (-0.811 + 0.584i)T \)
47 \( 1 + (0.984 - 0.174i)T \)
53 \( 1 + (-0.917 - 0.398i)T \)
59 \( 1 + (0.999 + 0.0390i)T \)
61 \( 1 + (-0.334 - 0.942i)T \)
67 \( 1 + (-0.279 + 0.960i)T \)
71 \( 1 + (-0.608 - 0.793i)T \)
73 \( 1 + (-0.917 + 0.398i)T \)
79 \( 1 + (0.184 + 0.982i)T \)
83 \( 1 + (-0.945 - 0.325i)T \)
89 \( 1 + (-0.279 + 0.960i)T \)
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

−21.70867631992125469823944620620, −20.6829944696217451502488296368, −19.768565536951037303237138969249, −19.164875169405487458439934151756, −18.55538522118348988641736934817, −17.540338038455280226716217501946, −16.88024195794184576168206530873, −16.228834238307780959793133310274, −15.39716363918826914936810628653, −14.581894593858125850710576524363, −12.99124938314626769705900509718, −12.22222964632319069721540986192, −11.93292991176715920682959374187, −11.20186580772067812069428789088, −10.23866676890301253621675301611, −9.40440496929038986921928235555, −8.834303042504832372733351660263, −7.555173213269392408731469177965, −6.90741572539702630889285924637, −5.66976120752179772194909685658, −4.612909543098701254496904954324, −3.80739306257266231195385613592, −2.95836884076550281786560364370, −1.51798557036899650457463338274, −0.473930315209564232661571588407, 0.63012109044278863508883093425, 1.08999642806200494222405476490, 3.016859434939230247458189527, 4.415756057936010786600867122377, 4.9422727297924700281654116430, 6.18353425062434080467228339682, 7.02723260629011744201597899748, 7.32638600918630771559508062146, 8.2619323854733444807545164880, 9.56528472958305355298753674505, 10.11730830642536830568685384416, 11.2082499585580343319360447368, 11.725517160969272370998197007, 12.73875803322992425417187932438, 13.78219732583427517665491434126, 14.55493863446821974963203271038, 15.55787794181826018506307800818, 16.34816754965290289053115311105, 16.74109985115151134729046505246, 17.488404765794418774909938987576, 18.352036757694620728641587111345, 19.218045602050808451490995413452, 19.6249127731903721865379589778, 20.462965492138637792851611145104, 22.1192556935764542991608049061

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