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Properties

Label 1-145-145.113-r0-0-0
Degree $1$
Conductor $145$
Sign $0.458 - 0.888i$
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  + (−0.900 + 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + (0.900 + 0.433i)6-s + (0.781 − 0.623i)7-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (0.974 − 0.222i)11-s − 12-s + (−0.974 + 0.222i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.222 − 0.974i)18-s + (−0.781 − 0.623i)19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + (0.900 + 0.433i)6-s + (0.781 − 0.623i)7-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (0.974 − 0.222i)11-s − 12-s + (−0.974 + 0.222i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.222 − 0.974i)18-s + (−0.781 − 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5295073976 - 0.3224856740i\)
\(L(\frac12)\) \(\approx\) \(0.5295073976 - 0.3224856740i\)
\(L(1)\) \(\approx\) \(0.6302438570 - 0.1508883743i\)
\(L(1)\) \(\approx\) \(0.6302438570 - 0.1508883743i\)

Euler product

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

−28.07299843531911394052361010228, −27.48261163037454888498490033273, −26.9860599339654766769693531654, −25.561226125347293015808014828159, −24.79715556282894659405786668182, −23.39146851134642108972960608149, −22.036090200067233036142679655, −21.476059980397897074224134619151, −20.52440820843314323281461726270, −19.41172335853840171800324612020, −18.26054419276329791752839592627, −17.24273129558214130273427765450, −16.747339734088311636721437608975, −15.36437682655632023345373939082, −14.57235803137606552350674416786, −12.33921531288686587572350970868, −11.83971490910767011608670638042, −10.73410470331245119519164293944, −9.726657571159155566771722488977, −8.84123661070921799903501950712, −7.52184652974083916072123942923, −6.05859004689931882549593883455, −4.67280997463149346323582232418, −3.2534346563560734169190005209, −1.528061970147000942827081350736, 0.86808099739638364707182801429, 2.168662022958665266669809830974, 4.6694725513293665198204269745, 5.98693737438461641959712086931, 7.08616163528238020402697825799, 7.82470520773259125621727954682, 9.1180858061657386522686020530, 10.54718217147191253786075880840, 11.3609215257595698161622393261, 12.42447488804830913614814093158, 14.07464038565500397224072234090, 14.78118858474973213928804204325, 16.540465300112468654301942337390, 17.06345570751033550484275575782, 17.8225946734321866610119359153, 19.01708100612411121874756709719, 19.62975338179276776083942005374, 20.91642722955639440573706657646, 22.38317838535544851430698412319, 23.538665617394719848607755111274, 24.266376725774578614646757534158, 24.9437727315170120072551844260, 26.121065671129369798111852513028, 27.34792225022716343598960574155, 27.7980187179389121872983482852

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