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Properties

Label 1-145-145.102-r0-0-0
Degree $1$
Conductor $145$
Sign $0.456 + 0.889i$
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  + (0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.222 − 0.974i)6-s + (0.433 − 0.900i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.781 + 0.623i)11-s + 12-s + (0.781 + 0.623i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s − 17-s + (−0.623 + 0.781i)18-s + (0.433 + 0.900i)19-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.222 − 0.974i)6-s + (0.433 − 0.900i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.781 + 0.623i)11-s + 12-s + (0.781 + 0.623i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s − 17-s + (−0.623 + 0.781i)18-s + (0.433 + 0.900i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.456 + 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7782007288 + 0.4753438718i\)
\(L(\frac12)\) \(\approx\) \(0.7782007288 + 0.4753438718i\)
\(L(1)\) \(\approx\) \(0.8417429482 + 0.3461250244i\)
\(L(1)\) \(\approx\) \(0.8417429482 + 0.3461250244i\)

Euler product

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

−28.32929106996042937157885693861, −27.35223889612381977414786870276, −26.663845733565612747951282830327, −24.87043651065836011713977079993, −23.88216466312675119936881989578, −22.76335247729011719272005710709, −22.02549496647839551188748242075, −21.34407497331040463454900022533, −20.3149462378468968051238282353, −19.04812896447525560921814872030, −18.048423383850332086748716979590, −17.34916069476856976147143110823, −15.809968655095721689006693194219, −14.90338644610700775531367107133, −13.49448605455386661188381574809, −12.398872450402085735346384220436, −11.34042365474527596238570591078, −10.92756537336497026662201831385, −9.42067511944267029071453938165, −8.61520453132203974482934846231, −6.374223100150774950576733054924, −5.35666016839663190538611640197, −4.30898805481511315182408818108, −2.90495809598156503657432262525, −1.10211906621036893563359722133, 1.3127596371058981307617114907, 4.002021719564727466353185099003, 4.87253805045028976590363046385, 6.31878234661445375318063248000, 6.99034278440470667317450266047, 8.09974123797410558007322179228, 9.563984219064035965494939017682, 10.98639170138409122350753917921, 12.080736878442670986859400438491, 13.30316888861554866227954381842, 14.05607710500489353850751199897, 15.36813265688943339003153329225, 16.54369158079134982367295696708, 17.2017260987975680009270345912, 18.00362484621285000653595598239, 19.07445562772711191920132749772, 20.61938896524757796471528108124, 21.88624330184427792417556315305, 22.89161840356132278188986717127, 23.41659129895103863541419306572, 24.4112252648295893711195062595, 25.166390694145237736435991270589, 26.48302161521025541439378177058, 27.30817589379406655613211309693, 28.24876332167439637856778141998

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