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Properties

Label 1-975-975.509-r0-0-0
Degree $1$
Conductor $975$
Sign $-0.988 + 0.150i$
Analytic cond. $4.52788$
Root an. cond. $4.52788$
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  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 + 0.5i)7-s + (0.587 − 0.809i)8-s + (−0.207 + 0.978i)11-s + (−0.309 − 0.951i)14-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.994 + 0.104i)19-s + (−0.913 − 0.406i)22-s + (0.978 + 0.207i)23-s + (0.994 − 0.104i)28-s + (0.104 + 0.994i)29-s + (−0.587 + 0.809i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 + 0.5i)7-s + (0.587 − 0.809i)8-s + (−0.207 + 0.978i)11-s + (−0.309 − 0.951i)14-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.994 + 0.104i)19-s + (−0.913 − 0.406i)22-s + (0.978 + 0.207i)23-s + (0.994 − 0.104i)28-s + (0.104 + 0.994i)29-s + (−0.587 + 0.809i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.988 + 0.150i$
Analytic conductor: \(4.52788\)
Root analytic conductor: \(4.52788\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 975,\ (0:\ ),\ -0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05612188974 + 0.7396919342i\)
\(L(\frac12)\) \(\approx\) \(0.05612188974 + 0.7396919342i\)
\(L(1)\) \(\approx\) \(0.6032857132 + 0.4505491981i\)
\(L(1)\) \(\approx\) \(0.6032857132 + 0.4505491981i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.207 + 0.978i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (0.994 + 0.104i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (0.743 - 0.669i)T \)
41 \( 1 + (-0.743 + 0.669i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.207 + 0.978i)T \)
97 \( 1 + (-0.406 + 0.913i)T \)
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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

−21.270311555382959794643423773776, −20.52806285875100885306315559699, −19.74032043278591404450724797261, −19.05613661433965355350035688121, −18.56315714087051838222078538960, −17.429802705731064377625732744234, −16.79063139762459145817840499494, −16.050785507861848792273116252692, −14.88528830726902731545898453718, −13.8543001652338627620829506657, −13.224168894781407854039298573045, −12.66164142179207120019225742091, −11.52079426136048037135878600154, −10.96392482448988365612271405209, −10.01352646077077941404568660428, −9.452356100219746406918705590232, −8.42918959223016173200669570903, −7.6814728898899402090788382765, −6.468277518514190469835549667822, −5.504337543390461636156073957258, −4.36679684317021248385100381858, −3.41891006574581108640599082701, −2.872515123656231498661952157415, −1.4929287028888741959271951548, −0.39238705442771528653492915291, 1.22962974835411661298258384453, 2.736201837765918959455132487797, 3.74789279927091188626850904095, 5.0807548798181485972535244230, 5.42281987468443690575590405770, 6.8470573629752356689220695712, 7.04945295570551470990193355488, 8.19118021272075404201630002101, 9.25904818506207247551567425322, 9.59786156079185789540954543608, 10.56165690818978434198007107414, 11.85933261916092929461974483373, 12.7325313015896124336362630904, 13.39267928527420004309444938657, 14.37447370414621205664969564634, 15.09870928940425940724413060249, 15.911773074526465244020745352076, 16.36783958650910190666094847437, 17.33715186933714957090196230747, 18.27749778524610892246806403545, 18.56259504676064918144758953850, 19.70509828784187862362582990753, 20.3156092078993990824072076767, 21.60391118857557759978789334251, 22.29279537556474095978278790673

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