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Properties

Label 1-145-145.42-r1-0-0
Degree $1$
Conductor $145$
Sign $0.0964 - 0.995i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + (0.781 − 0.623i)2-s + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (0.623 − 0.781i)6-s + (0.974 − 0.222i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s − i·12-s + (−0.433 + 0.900i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (0.433 − 0.900i)18-s + (−0.222 + 0.974i)19-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (0.623 − 0.781i)6-s + (0.974 − 0.222i)7-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s − i·12-s + (−0.433 + 0.900i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (0.433 − 0.900i)18-s + (−0.222 + 0.974i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0964 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0964 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.0964 - 0.995i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.0964 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.182622359 - 2.889178298i\)
\(L(\frac12)\) \(\approx\) \(3.182622359 - 2.889178298i\)
\(L(1)\) \(\approx\) \(2.129914280 - 1.186914489i\)
\(L(1)\) \(\approx\) \(2.129914280 - 1.186914489i\)

Euler product

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

−27.84696214669218808830122868688, −27.04629322671967640724740687348, −26.06245165006041611724295321161, −25.08018232008923586951060059837, −24.44970998665649663242405683177, −23.56445518128713655733770553940, −21.85149801497703219496859296115, −21.70713123966422529451156626765, −20.3632994909787413584537050196, −19.585902341808353409829112655538, −17.97170124541379368048400297816, −17.02688636060294487034701642336, −15.67204771034692324434355807025, −14.89149608193713459335647211634, −14.203729715451572747517664963020, −13.17786273074432560038408873895, −12.03050848555598312012071771874, −10.71820130593406036652374513448, −8.9941635250454361256532362937, −8.18972989253688039409905568616, −7.18394749787511677995857085934, −5.6514942207055493507590448357, −4.42873678681256486495600947962, −3.369855624906915659376627577804, −1.96463587434626345411935806783, 1.39291410554275667477791511146, 2.33473834792946922395886670325, 3.89487127258943915950511125855, 4.69024537522718859105306798290, 6.47457163763376990608054236724, 7.65159170034340288376907159192, 9.1131442337500422199409912319, 10.060302050619780924493823922249, 11.544739125215584502976650468414, 12.30886993888389423083733648971, 13.67394357062246689997886322039, 14.393803175146370364796126987977, 14.956028194965640805228688188773, 16.48710095682533231903603705716, 18.12692371803548311414458624716, 19.00996396177014953809051493483, 20.11009116036855575751012417617, 20.67210295309530710679443764300, 21.62220481295054817922081881909, 22.72397121328423554419910279922, 24.02627437702630001060993594375, 24.52482816992827881258425546490, 25.57419847283103248491200105756, 27.03135731625794284405058597254, 27.64566339204277050978791332427

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