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Properties

Label 1-975-975.839-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: $\chi_{975} (839, \cdot )$
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

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

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