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Properties

Label 1-953-953.134-r0-0-0
Degree $1$
Conductor $953$
Sign $-0.415 + 0.909i$
Analytic cond. $4.42571$
Root an. cond. $4.42571$
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  + (0.932 − 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 − 0.673i)4-s + (0.445 + 0.895i)5-s + (−0.602 − 0.798i)6-s + (−0.982 − 0.183i)7-s + (0.445 − 0.895i)8-s + (−0.850 + 0.526i)9-s + (0.739 + 0.673i)10-s + (−0.982 − 0.183i)11-s + (−0.850 − 0.526i)12-s + (−0.850 + 0.526i)13-s + (−0.982 + 0.183i)14-s + (0.739 − 0.673i)15-s + (0.0922 − 0.995i)16-s + (−0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 − 0.673i)4-s + (0.445 + 0.895i)5-s + (−0.602 − 0.798i)6-s + (−0.982 − 0.183i)7-s + (0.445 − 0.895i)8-s + (−0.850 + 0.526i)9-s + (0.739 + 0.673i)10-s + (−0.982 − 0.183i)11-s + (−0.850 − 0.526i)12-s + (−0.850 + 0.526i)13-s + (−0.982 + 0.183i)14-s + (0.739 − 0.673i)15-s + (0.0922 − 0.995i)16-s + (−0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $-0.415 + 0.909i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ -0.415 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003687747320 + 0.0005738766014i\)
\(L(\frac12)\) \(\approx\) \(0.0003687747320 + 0.0005738766014i\)
\(L(1)\) \(\approx\) \(0.9820794975 - 0.4295937009i\)
\(L(1)\) \(\approx\) \(0.9820794975 - 0.4295937009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad953 \( 1 \)
good2 \( 1 + (0.932 - 0.361i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (0.445 + 0.895i)T \)
7 \( 1 + (-0.982 - 0.183i)T \)
11 \( 1 + (-0.982 - 0.183i)T \)
13 \( 1 + (-0.850 + 0.526i)T \)
17 \( 1 + (-0.602 - 0.798i)T \)
19 \( 1 + (-0.850 + 0.526i)T \)
23 \( 1 + (0.0922 + 0.995i)T \)
29 \( 1 + (-0.602 - 0.798i)T \)
31 \( 1 + (-0.850 + 0.526i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (0.739 - 0.673i)T \)
43 \( 1 + (0.932 + 0.361i)T \)
47 \( 1 + (-0.602 + 0.798i)T \)
53 \( 1 + (-0.602 - 0.798i)T \)
59 \( 1 + (-0.602 - 0.798i)T \)
61 \( 1 + (-0.982 + 0.183i)T \)
67 \( 1 + (-0.850 + 0.526i)T \)
71 \( 1 + (-0.850 + 0.526i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.850 - 0.526i)T \)
83 \( 1 + (-0.982 + 0.183i)T \)
89 \( 1 + (0.445 - 0.895i)T \)
97 \( 1 + (-0.982 - 0.183i)T \)
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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

−21.723952962881282939430838434756, −20.9918724951156307506453760029, −20.15028041823070957143355006267, −19.71119342414491092307929395361, −18.093553136711769337762522380611, −17.138051308466557903553761283189, −16.62741361688439520978895156903, −15.96988285032430530037362650606, −15.1923829055827415182880625003, −14.65367702732258661716463624383, −13.32256780732680733204840805314, −12.778911031966494727775271387881, −12.31120834875491614053043485209, −10.94512242821667239740288506033, −10.35976143113780250341175836123, −9.30711485993380336856896727630, −8.57864679313769325994093715138, −7.44703461731500882858970445475, −6.17316438463859662848685010560, −5.72478607379183367019583946735, −4.75051433010716677387924803595, −4.24155562580813601262905516078, −2.99371046016392300130391181795, −2.25787518023855633492152248168, −0.00018771028973094745002970846, 1.76686596761358658230731263663, 2.57086571555506798455036434294, 3.16498755081532811978304775548, 4.50972503028156776220757557708, 5.73320353615578894835791686271, 6.15236371904179634351433202306, 7.16381163703342744762266573234, 7.51408858912310439587792121513, 9.31253944170336748135506812727, 10.138277928360579241314819967376, 11.00759018820196576684139852241, 11.61298631347619952506383608713, 12.77018403411779344855150967319, 13.08917547546570768894099954080, 13.92926454791336970989753162232, 14.54738651633464513079384080012, 15.58441946030001985796445289263, 16.41446747092688789877831241144, 17.39831622631340506942678896625, 18.35004204113003650921933961563, 19.124915401777999476023394599445, 19.43018335385655882225192873000, 20.51362720785338907785850483034, 21.48847812018450904746973345154, 22.1913234664589029955633719522

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