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Properties

Label 1-145-145.33-r1-0-0
Degree $1$
Conductor $145$
Sign $0.633 + 0.773i$
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  + (0.433 − 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 + 0.433i)6-s + (−0.781 − 0.623i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.222 − 0.974i)11-s + i·12-s + (−0.974 − 0.222i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.974 + 0.222i)18-s + (0.623 + 0.781i)19-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 + 0.433i)6-s + (−0.781 − 0.623i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (0.222 − 0.974i)11-s + i·12-s + (−0.974 − 0.222i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.974 + 0.222i)18-s + (0.623 + 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.633 + 0.773i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.633 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009716831788 + 0.004601159398i\)
\(L(\frac12)\) \(\approx\) \(0.009716831788 + 0.004601159398i\)
\(L(1)\) \(\approx\) \(0.4906362558 - 0.4865050768i\)
\(L(1)\) \(\approx\) \(0.4906362558 - 0.4865050768i\)

Euler product

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

−28.70234779836877061870675479796, −27.72986242727446661206131740715, −26.78424308953872182950290873729, −25.845798626095399149836816256659, −24.88851243463673382134900368875, −23.82281338802897681653696666473, −22.7285594739027890429320055117, −22.26226927866542451826051460791, −21.3979559758096627753479261052, −20.028114891763595216399709730756, −18.418234918171749143136405980478, −17.51404847859525285977147513404, −16.61896110587079750170353897659, −15.61286301242375240864358176722, −15.07599663208119695024502751352, −13.648021818613856584188681313468, −12.32309014252069134310194030355, −11.74879129438404439170670645138, −9.74202378329453266504130244915, −9.30308639477250047173703720243, −7.40360522765604004305049317978, −6.47279468571137071880931335899, −5.30178628167273139940377933761, −4.45145652547422099192836532732, −2.95909275191779879421904931490, 0.00430465762664938163049105682, 1.25310543310961030175202179328, 2.89744758884741724910380061243, 4.28032879106424170199741412766, 5.68144041748174983267696062548, 6.57286513507003141368954505594, 8.16001219428952536816080365850, 9.8834047898141994348352214283, 10.63043530818062127399281384405, 11.79886118037351821162674984678, 12.646109661560566591044236964980, 13.50205676001807605401546662700, 14.52605375653769431408703060413, 16.23058905048970901576897349212, 17.15963511295750049245376295797, 18.35939839596275537328976695261, 19.30427403054739655916657600275, 19.91568778131540262995667602633, 21.41972720907204654887932344914, 22.30998908723424259853111197635, 22.96899543162974540109824582332, 23.99619076242072264925481013788, 24.74159933598765040995754486841, 26.49035889946943191573299066232, 27.36055665493546925905783786045

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