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Properties

Label 1-145-145.129-r0-0-0
Degree $1$
Conductor $145$
Sign $0.441 + 0.897i$
Analytic cond. $0.673377$
Root an. cond. $0.673377$
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  + (0.623 + 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + 12-s + (0.900 + 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.900 − 0.433i)16-s + 17-s + (−0.900 − 0.433i)18-s + (0.222 − 0.974i)19-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + 12-s + (0.900 + 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.900 − 0.433i)16-s + 17-s + (−0.900 − 0.433i)18-s + (0.222 − 0.974i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.176582629 + 0.7325906315i\)
\(L(\frac12)\) \(\approx\) \(1.176582629 + 0.7325906315i\)
\(L(1)\) \(\approx\) \(1.233397880 + 0.4601617061i\)
\(L(1)\) \(\approx\) \(1.233397880 + 0.4601617061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + T \)
19 \( 1 + (0.222 - 0.974i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 - T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (0.900 - 0.433i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−27.920002311337980884886504574413, −27.42779731089551481594213816283, −26.434965856586298164636875705318, −25.03993810341640231365424268302, −23.65645511535626488827184031844, −22.92568734396000975638150537317, −22.131890168801051999446800763890, −20.97218198657379202481926824858, −20.48331901330737970542893948321, −19.454803345529655657837309447859, −18.11224515201867526334971199056, −16.8100608824334766674980724318, −15.90430460135011637118282917061, −14.423845119493104697343891366827, −14.086415072375177261748387003717, −12.51588621345646918124531190477, −11.39009309588318585012862467126, −10.56144785442945017705112352703, −9.75421074172892329736750211593, −8.39837265298785648351564901262, −6.37681810635626447192919791635, −5.29804418836905265226863108155, −3.96649689265070568569931487081, −3.4043077253417551694351944224, −1.21455985776326658252563532446, 1.88478346650419363814312458, 3.476014505343276000727260227290, 5.16265338177955835845346795533, 6.10735425741239903935702992364, 7.06629884529485073834327272600, 8.23044045032822948633815046253, 9.19888846227728439172852122447, 11.541697993432677104216515728068, 12.03293414535799740302606638589, 13.19373641424856430734407023569, 14.1215116184580099166471095744, 15.1059662686411578117074913083, 16.28697532573824583032347969483, 17.383754389526900038015024978041, 18.17962255189699160541783724604, 19.18537793983064662823184875756, 20.62818033927817179923093375218, 21.862459636490082125477305746921, 22.62442798301608423553906917514, 23.72468797016686910670549115862, 24.33045040158551000371131647387, 25.48016731579069118918573730723, 25.745144538128374586773355808, 27.55326541083701298845658403020, 28.34782998146016247711751317915

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