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Properties

Label 1-363-363.149-r0-0-0
Degree $1$
Conductor $363$
Sign $0.874 + 0.484i$
Analytic cond. $1.68576$
Root an. cond. $1.68576$
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  + (−0.870 − 0.491i)2-s + (0.516 + 0.856i)4-s + (−0.941 − 0.336i)5-s + (−0.897 − 0.441i)7-s + (−0.0285 − 0.999i)8-s + (0.654 + 0.755i)10-s + (−0.0855 + 0.996i)13-s + (0.564 + 0.825i)14-s + (−0.466 + 0.884i)16-s + (−0.998 − 0.0570i)17-s + (0.254 − 0.967i)19-s + (−0.198 − 0.980i)20-s + (0.142 + 0.989i)23-s + (0.774 + 0.633i)25-s + (0.564 − 0.825i)26-s + ⋯
L(s)  = 1  + (−0.870 − 0.491i)2-s + (0.516 + 0.856i)4-s + (−0.941 − 0.336i)5-s + (−0.897 − 0.441i)7-s + (−0.0285 − 0.999i)8-s + (0.654 + 0.755i)10-s + (−0.0855 + 0.996i)13-s + (0.564 + 0.825i)14-s + (−0.466 + 0.884i)16-s + (−0.998 − 0.0570i)17-s + (0.254 − 0.967i)19-s + (−0.198 − 0.980i)20-s + (0.142 + 0.989i)23-s + (0.774 + 0.633i)25-s + (0.564 − 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ 0.874 + 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4415786652 + 0.1142339855i\)
\(L(\frac12)\) \(\approx\) \(0.4415786652 + 0.1142339855i\)
\(L(1)\) \(\approx\) \(0.5201189873 - 0.06946753400i\)
\(L(1)\) \(\approx\) \(0.5201189873 - 0.06946753400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.870 - 0.491i)T \)
5 \( 1 + (-0.941 - 0.336i)T \)
7 \( 1 + (-0.897 - 0.441i)T \)
13 \( 1 + (-0.0855 + 0.996i)T \)
17 \( 1 + (-0.998 - 0.0570i)T \)
19 \( 1 + (0.254 - 0.967i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.774 - 0.633i)T \)
31 \( 1 + (0.974 + 0.226i)T \)
37 \( 1 + (-0.921 + 0.389i)T \)
41 \( 1 + (-0.736 + 0.676i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.993 - 0.113i)T \)
53 \( 1 + (0.466 + 0.884i)T \)
59 \( 1 + (0.736 + 0.676i)T \)
61 \( 1 + (0.870 - 0.491i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.362 + 0.931i)T \)
73 \( 1 + (0.985 + 0.170i)T \)
79 \( 1 + (-0.610 + 0.791i)T \)
83 \( 1 + (0.696 - 0.717i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.941 - 0.336i)T \)
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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

−24.75967533389296386574071326814, −23.96947664234312439463313377133, −22.703881395711467656419643398656, −22.56520194752041944564096935678, −20.80592874161585318472498753658, −19.88195815555340515901947059942, −19.27137111438448088485281313083, −18.48632121670111617191301150254, −17.655278600338159955677489005026, −16.46892011495939172761895067520, −15.757403453990792755312887288945, −15.19892795447762080352468719090, −14.17888849102717339237000659845, −12.724735014712335613212326643186, −11.85740432933095726360601455651, −10.69147093674323452170617483758, −10.04854319954401335098687527260, −8.77193502548345974259404356011, −8.12441029944396395550698328897, −6.98961169982996581393843717228, −6.278282659367791533524157189359, −5.035064510168372032283515370438, −3.48718706748536911283267659994, −2.379056673260876815675004414153, −0.46783086456534618296978450132, 0.9929081923985439259783134898, 2.599511979114639157013972797004, 3.68823667619404315372482630409, 4.604878908610535021997883309843, 6.61886607136212269356069306999, 7.17539695001792841633011234224, 8.370386415267240628404926843572, 9.18484430646020485333557207196, 10.05963011350934938859298336286, 11.27131652818685089911002705635, 11.79696090606520019860207593034, 12.88977864077107594657693583798, 13.70581916073717244060526633131, 15.51264367304075404019832446005, 15.89188272253947201230357359846, 16.87932698037390144495730366655, 17.650855344170909938310252615317, 18.92388440629146299756318816172, 19.51891928214248828086258251359, 20.008011365046168841321918205241, 21.08071563278330864314273228549, 22.02714988299144210305712825469, 22.99752087033643633974863606450, 23.9806533987389993530608886955, 24.84281918204419578867090087563

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