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Properties

Label 1-145-145.49-r0-0-0
Degree $1$
Conductor $145$
Sign $0.128 - 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.128 - 0.991i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.128 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5283904289 - 0.4643430932i\)
\(L(\frac12)\) \(\approx\) \(0.5283904289 - 0.4643430932i\)
\(L(1)\) \(\approx\) \(0.7268306893 - 0.1919772141i\)
\(L(1)\) \(\approx\) \(0.7268306893 - 0.1919772141i\)

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

−28.42993690434983436054601065125, −27.562821465258759011150674739, −26.65320778795882874762843934886, −25.872227269064260744935882389467, −24.93359669600011781567019450236, −23.26190761373841646574935132524, −22.033492313695805910303424475441, −21.34533627839123870266902901850, −20.70982795912479672501615771994, −19.54011454260497885016857382042, −18.534353068393804533749965175904, −17.630185987601800929801607661226, −16.19711224887297133206486824542, −15.68742395674225610508936132819, −14.1757625104475267751557338466, −12.97495966900626017903822437139, −11.59115603357842104286380182114, −10.87654800560407029613546896049, −9.74883212964453398665736168421, −8.74187932240672495719230073270, −8.049970610407453929256567220191, −5.910880354541151273323698258878, −4.48808589866343090969546155527, −3.25160225345559796985255635189, −2.079345067882472119038615344582, 0.73475961966970179134532702205, 2.31826943645423395669771680452, 4.4621661529749354543305285602, 6.00587702115568388921559071386, 7.01683826362336735252348391887, 7.89196599556513121706416900895, 8.78647490167904093138411490100, 10.30052517157561856724024957642, 11.25935957408459739774683376576, 13.11973547553402707761095596758, 13.63303551021645571475174289687, 14.87866075398225969026054088714, 15.920286473433190602561875055893, 17.242284793635746873965478240702, 17.90205294534059457977177205, 18.72322998141409529306957470975, 19.96420074290775081490229176870, 20.507309031549710780993319182741, 22.58348468867687232991770705393, 23.55920837294733081113129191131, 24.02354047871584940059065994142, 25.12760092461751341973584153350, 26.09409459014991835405680126960, 26.60240972174548646349696696181, 28.10799504719426432334743228843

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