Properties

Label 1-75-75.41-r1-0-0
Degree $1$
Conductor $75$
Sign $-0.425 - 0.904i$
Analytic cond. $8.05986$
Root an. cond. $8.05986$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + 7-s + (0.809 + 0.587i)8-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.809 + 0.587i)22-s + (−0.309 − 0.951i)23-s − 26-s + (−0.809 + 0.587i)28-s + (0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + 7-s + (0.809 + 0.587i)8-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.809 + 0.587i)22-s + (−0.309 − 0.951i)23-s − 26-s + (−0.809 + 0.587i)28-s + (0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(8.05986\)
Root analytic conductor: \(8.05986\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 75,\ (1:\ ),\ -0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6984830320 - 1.100633142i\)
\(L(\frac12)\) \(\approx\) \(0.6984830320 - 1.100633142i\)
\(L(1)\) \(\approx\) \(0.8038198119 - 0.5313852746i\)
\(L(1)\) \(\approx\) \(0.8038198119 - 0.5313852746i\)

Euler product

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

−31.49461867665168233266537038627, −30.77333467250866037898057373199, −29.139972881805159378956237587511, −27.83966497197418379941909681980, −27.256893130984413024278621351109, −25.87873576502764176000989947005, −25.16770834867870530744392903001, −23.79422234864606082760735629845, −23.28981127790135835452015518693, −21.73247141063655223611652210330, −20.52051260730950143183232816822, −18.98061685966878350284240630609, −18.013857161006477883342426707174, −17.05423629228625705601275300282, −15.854768362762990886654946889111, −14.68458710416845162938030967038, −13.86538073120122508518109999450, −12.21779050712380242841217324863, −10.61991483697749299338923650917, −9.31006377719096854250487436531, −8.04742286217164856112177213261, −7.00540406980156394341900099318, −5.438002765557123987743311529824, −4.27725795328239511451686177981, −1.59180805808943566560797878180, 0.79665559333047816359873096088, 2.52268401807257399423458949706, 4.08617490691351870312083883907, 5.59607929199883665526562636617, 7.89150456082897589899747235103, 8.66878304157677214152472568099, 10.383596710847931295165537200711, 11.13178690319498464398024090224, 12.44482960597555934961094500124, 13.60608017957767904461359237673, 14.85301453389654748844893519757, 16.59874073608971153277663092792, 17.73595296366199986355140150445, 18.64695299859915888244813547222, 19.82412418922929475538881023903, 20.98068916570113529166618519568, 21.64866655527811080976967066645, 23.02466875441251342269817981203, 24.16655332863320378961234398865, 25.63120067221417547056698929839, 26.82797573985535928531677345761, 27.66316779715092755872688061575, 28.56317518667566528340895554967, 29.95747276014585730098602878766, 30.39962159591193327458838845072

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