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Properties

Label 1-953-953.604-r0-0-0
Degree $1$
Conductor $953$
Sign $-0.383 - 0.923i$
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  + (0.0922 + 0.995i)2-s + (0.445 + 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.273 + 0.961i)5-s + (−0.850 + 0.526i)6-s + (0.739 − 0.673i)7-s + (−0.273 − 0.961i)8-s + (−0.602 + 0.798i)9-s + (−0.982 − 0.183i)10-s + (0.739 − 0.673i)11-s + (−0.602 − 0.798i)12-s + (−0.602 + 0.798i)13-s + (0.739 + 0.673i)14-s + (−0.982 + 0.183i)15-s + (0.932 − 0.361i)16-s + (−0.850 + 0.526i)17-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (0.445 + 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.273 + 0.961i)5-s + (−0.850 + 0.526i)6-s + (0.739 − 0.673i)7-s + (−0.273 − 0.961i)8-s + (−0.602 + 0.798i)9-s + (−0.982 − 0.183i)10-s + (0.739 − 0.673i)11-s + (−0.602 − 0.798i)12-s + (−0.602 + 0.798i)13-s + (0.739 + 0.673i)14-s + (−0.982 + 0.183i)15-s + (0.932 − 0.361i)16-s + (−0.850 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $-0.383 - 0.923i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ -0.383 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5454532894 + 0.8175320091i\)
\(L(\frac12)\) \(\approx\) \(-0.5454532894 + 0.8175320091i\)
\(L(1)\) \(\approx\) \(0.4905995742 + 0.8523875945i\)
\(L(1)\) \(\approx\) \(0.4905995742 + 0.8523875945i\)

Euler product

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

−20.829941099464783790023977374, −20.45451387338659416289606381011, −19.75708533275742715161362966746, −19.12071979158431251203151550680, −18.20885599568751377492979788698, −17.490042775804351069771415883222, −17.00830459306878280700181689216, −15.11968720295850781512846676250, −14.979646458668942400183577811886, −13.77376642262377595713930475536, −12.88834402057834796964545183325, −12.59690629147230951661084374933, −11.62179536962430572161458295497, −11.21454898417402079056924709488, −9.57210081631857200108557511542, −9.093650190142191324060012848991, −8.32419301497813800704570308675, −7.57845282948618304127194454971, −6.27880166261721809621720639162, −5.01975830977342698577858369607, −4.56546211509217816135238391304, −3.235915706481134456965975376983, −2.209480464823895327316808043412, −1.587737997254702105806239257, −0.37878391278819163938722609875, 1.84663985778129299478314327504, 3.396573803855482418110786550758, 3.942948308896228236647051911393, 4.72046032087966061020842958357, 5.79211530418655889131862325091, 6.86374040705197789617662368902, 7.46175375723477699490963009242, 8.52157120343173951851900261064, 9.060384250564864507568883545370, 10.18390723244812554375500298893, 10.882426920750705428222675826001, 11.68178517237871886367920179876, 13.147603846742226564803323942277, 14.10834571874373132558370965456, 14.51063261261975510421555314998, 14.979283899337524086860326771594, 15.92153203892804603683792678865, 16.857317331989167568831518832173, 17.175164390487760670372002249377, 18.32707564292290862152165385962, 19.21634765052118510949445077001, 19.78927776652254819595958567852, 21.05160834190317706846650294313, 21.78899807769856126199151268064, 22.19457508171799097852081227333

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