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Properties

Label 1-31e2-961.112-r0-0-0
Degree $1$
Conductor $961$
Sign $0.249 + 0.968i$
Analytic cond. $4.46286$
Root an. cond. $4.46286$
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  + (0.111 + 0.993i)2-s + (−0.776 + 0.630i)3-s + (−0.975 + 0.221i)4-s + (0.217 + 0.975i)5-s + (−0.713 − 0.701i)6-s + (−0.968 − 0.247i)7-s + (−0.328 − 0.944i)8-s + (0.204 − 0.978i)9-s + (−0.945 + 0.325i)10-s + (−0.880 + 0.473i)11-s + (0.617 − 0.786i)12-s + (0.952 − 0.305i)13-s + (0.138 − 0.990i)14-s + (−0.784 − 0.620i)15-s + (0.902 − 0.431i)16-s + (−0.958 − 0.286i)17-s + ⋯
L(s)  = 1  + (0.111 + 0.993i)2-s + (−0.776 + 0.630i)3-s + (−0.975 + 0.221i)4-s + (0.217 + 0.975i)5-s + (−0.713 − 0.701i)6-s + (−0.968 − 0.247i)7-s + (−0.328 − 0.944i)8-s + (0.204 − 0.978i)9-s + (−0.945 + 0.325i)10-s + (−0.880 + 0.473i)11-s + (0.617 − 0.786i)12-s + (0.952 − 0.305i)13-s + (0.138 − 0.990i)14-s + (−0.784 − 0.620i)15-s + (0.902 − 0.431i)16-s + (−0.958 − 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ 0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5529921408 + 0.4285298051i\)
\(L(\frac12)\) \(\approx\) \(0.5529921408 + 0.4285298051i\)
\(L(1)\) \(\approx\) \(0.5236449304 + 0.4420012925i\)
\(L(1)\) \(\approx\) \(0.5236449304 + 0.4420012925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.111 + 0.993i)T \)
3 \( 1 + (-0.776 + 0.630i)T \)
5 \( 1 + (0.217 + 0.975i)T \)
7 \( 1 + (-0.968 - 0.247i)T \)
11 \( 1 + (-0.880 + 0.473i)T \)
13 \( 1 + (0.952 - 0.305i)T \)
17 \( 1 + (-0.958 - 0.286i)T \)
19 \( 1 + (0.0573 - 0.998i)T \)
23 \( 1 + (0.947 + 0.318i)T \)
29 \( 1 + (-0.579 - 0.814i)T \)
37 \( 1 + (0.409 - 0.912i)T \)
41 \( 1 + (0.973 - 0.227i)T \)
43 \( 1 + (0.956 - 0.292i)T \)
47 \( 1 + (-0.0101 + 0.999i)T \)
53 \( 1 + (-0.987 - 0.154i)T \)
59 \( 1 + (-0.740 - 0.671i)T \)
61 \( 1 + (-0.440 - 0.897i)T \)
67 \( 1 + (-0.184 + 0.982i)T \)
71 \( 1 + (0.967 + 0.253i)T \)
73 \( 1 + (0.828 - 0.560i)T \)
79 \( 1 + (0.828 + 0.560i)T \)
83 \( 1 + (0.805 - 0.593i)T \)
89 \( 1 + (-0.832 - 0.554i)T \)
97 \( 1 + (-0.511 + 0.859i)T \)
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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

−21.50519345651277441093443917779, −20.90495437485259658759974018707, −19.99752692696890972492722938667, −19.20852250254973729355250373460, −18.50627350649024435214587356362, −17.987032492906634633922408735745, −16.82707502966937280281646441078, −16.39685951864825163095525791919, −15.40385317843269770897904156747, −13.8472320012911067872502149812, −13.253158965612462647163169045188, −12.766375739126336268008350311514, −12.17657007973079978542812473102, −11.09882780196619685179914816849, −10.590629443352859807939150986615, −9.48061733959250451848101799352, −8.72976947890626594416369806675, −7.90317211538589651510274445514, −6.40363073052237143589724904579, −5.76108278554061202233504705766, −4.96289882995295582980793623812, −3.95416021908200456201201596164, −2.7773144074793103894732094989, −1.72519633662266429048434834389, −0.796474356312103430956044905161, 0.479954291965145623544031328426, 2.71965910546711740274734782001, 3.63610430485086238631647563755, 4.52729871446927307923872512301, 5.55977462074155368859128652959, 6.263685301697321453234430422675, 6.90785747029624742487680515429, 7.69062653674561181768117781743, 9.236091521461762850653849097026, 9.55461690910464026691462429079, 10.735597774114723313099959476194, 11.09620614805446737420275032741, 12.663697618184729595324331740279, 13.19742335042067984565846407979, 14.05624006773837846259476548487, 15.28318258465125179671526306061, 15.551930971804749093542766302114, 16.147026822730403692973609565097, 17.29824055137470948119722728328, 17.73834264034024260455476577721, 18.4694649499817728137724165896, 19.296364599432337437480092510820, 20.61567661874958169541172098023, 21.46808666476860580832183423728, 22.25705128086813978487817728737

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