Properties

Label 1-968-968.107-r0-0-0
Degree $1$
Conductor $968$
Sign $-0.775 + 0.631i$
Analytic cond. $4.49537$
Root an. cond. $4.49537$
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  + (−0.809 + 0.587i)3-s + (0.736 − 0.676i)5-s + (−0.998 − 0.0570i)7-s + (0.309 − 0.951i)9-s + (−0.564 + 0.825i)13-s + (−0.198 + 0.980i)15-s + (0.921 + 0.389i)17-s + (−0.974 − 0.226i)19-s + (0.841 − 0.540i)21-s + (−0.841 − 0.540i)23-s + (0.0855 − 0.996i)25-s + (0.309 + 0.951i)27-s + (0.0855 + 0.996i)29-s + (0.0285 − 0.999i)31-s + (−0.774 + 0.633i)35-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)3-s + (0.736 − 0.676i)5-s + (−0.998 − 0.0570i)7-s + (0.309 − 0.951i)9-s + (−0.564 + 0.825i)13-s + (−0.198 + 0.980i)15-s + (0.921 + 0.389i)17-s + (−0.974 − 0.226i)19-s + (0.841 − 0.540i)21-s + (−0.841 − 0.540i)23-s + (0.0855 − 0.996i)25-s + (0.309 + 0.951i)27-s + (0.0855 + 0.996i)29-s + (0.0285 − 0.999i)31-s + (−0.774 + 0.633i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(4.49537\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (0:\ ),\ -0.775 + 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1301574670 + 0.3662935319i\)
\(L(\frac12)\) \(\approx\) \(0.1301574670 + 0.3662935319i\)
\(L(1)\) \(\approx\) \(0.6716101809 + 0.1030925331i\)
\(L(1)\) \(\approx\) \(0.6716101809 + 0.1030925331i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.736 - 0.676i)T \)
7 \( 1 + (-0.998 - 0.0570i)T \)
13 \( 1 + (-0.564 + 0.825i)T \)
17 \( 1 + (0.921 + 0.389i)T \)
19 \( 1 + (-0.974 - 0.226i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.0855 + 0.996i)T \)
31 \( 1 + (0.0285 - 0.999i)T \)
37 \( 1 + (-0.941 - 0.336i)T \)
41 \( 1 + (0.466 + 0.884i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (-0.696 - 0.717i)T \)
53 \( 1 + (0.254 + 0.967i)T \)
59 \( 1 + (-0.466 + 0.884i)T \)
61 \( 1 + (0.897 - 0.441i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.516 + 0.856i)T \)
73 \( 1 + (0.362 + 0.931i)T \)
79 \( 1 + (0.993 - 0.113i)T \)
83 \( 1 + (-0.774 - 0.633i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.736 - 0.676i)T \)
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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

−21.65852443535731969783711348154, −20.81141221820094533644879630623, −19.436356744711498730887380281600, −19.14777798300234462204159170226, −18.20555816074092824398835634904, −17.539730639483584027951018642395, −16.905549720849740249692725506874, −16.02770297199621905084349921477, −15.15574092306461889708673092462, −14.060202856248660711756278661710, −13.4366354689009124170031454275, −12.52977625787658688708231924606, −12.03895059632590389183349120964, −10.813104032232709404874269191740, −10.18094032403457682375287952660, −9.611962803589497357211477477914, −8.18524606492236564368161555959, −7.246183945147793620397288245242, −6.529348336011412300686559495026, −5.815339362928097590740485812406, −5.134342879138518890151497923329, −3.618466115380109502729079836118, −2.64417105470269159505365415586, −1.705874725130615918352560800506, −0.19001240629978463947791021186, 1.25049403653211341525457265681, 2.51615203556978279313209515078, 3.831319971489543680409073826303, 4.57578641316188873207278539572, 5.56241698012070504352654268988, 6.24032048970139346340293858705, 6.9664445786036820426801048279, 8.43851735594921328424747621968, 9.35305370300043698583849261302, 9.91901953723280318626731621067, 10.55231665126154645801161399117, 11.740889555299153356616185045526, 12.50202657951343957955035514055, 13.015190645687374933478082098518, 14.16634537510189179337999591738, 14.96883170748245802814456990815, 16.09972257384412644228763393130, 16.59121266330405078664573228887, 17.04365170261308634871287841852, 17.97199178669723558627808366650, 18.872092318483875277436272519674, 19.77634070931992482125981731870, 20.6386820890961575680956210659, 21.580778278172586604168802368650, 21.7812474872649721753257588530

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