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Properties

Label 1-369-369.155-r1-0-0
Degree $1$
Conductor $369$
Sign $0.489 + 0.871i$
Analytic cond. $39.6545$
Root an. cond. $39.6545$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 8-s + 10-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)13-s + (0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s − i·17-s + i·19-s + (−0.5 − 0.866i)20-s + (−0.866 − 0.5i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 8-s + 10-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)13-s + (0.866 + 0.5i)14-s + (−0.5 − 0.866i)16-s − i·17-s + i·19-s + (−0.5 − 0.866i)20-s + (−0.866 − 0.5i)22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(39.6545\)
Root analytic conductor: \(39.6545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 369,\ (1:\ ),\ 0.489 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8470722961 + 0.4956844221i\)
\(L(\frac12)\) \(\approx\) \(0.8470722961 + 0.4956844221i\)
\(L(1)\) \(\approx\) \(0.7305430903 - 0.03528032107i\)
\(L(1)\) \(\approx\) \(0.7305430903 - 0.03528032107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−24.2575817015011119670096123444, −23.52991096592524672911384665062, −22.91890371807321989747280131505, −21.88105074051563035089848097226, −20.392672177176364369697792154884, −19.70551826874596033016704233171, −19.1679440708454721285754863702, −17.784036387928414349044637555158, −17.1062597290328459332706024449, −16.30798211792469228563913512832, −15.574124586864025847574952275595, −14.7581074871511252267127797553, −13.3667374379116132358629816302, −12.93578563460360211005138520545, −11.50830080878319156673523601690, −10.37316763377140733735449847033, −9.33891999077290565736272834853, −8.68868507891827366865133670323, −7.61382965405061100107856340309, −6.69823554843005023531045086795, −5.73471320733862044040322419747, −4.50061361830620621657857534356, −3.62032263858005172060414503717, −1.41897561790158561796667843979, −0.430094496271257406401337274467, 1.028895027640457682077342652654, 2.60887738414981361824319620827, 3.36556903492908687216893788703, 4.29777934089629819806977620351, 6.1340638307550638124064595932, 6.98235102526743190665708671310, 8.24832393876200274056096109466, 9.12113111459851099464215185331, 10.02001570788889057261890416261, 11.062217804981687280927092923966, 11.735016139732489610588646362778, 12.5836905463323656703844129872, 13.75998338488157358796011879871, 14.56652610709913645516771621287, 16.059166982358014031314318480970, 16.45115860446712133733687382997, 17.89774993383196423916802680182, 18.68405574172027727174774382420, 19.14631697075871259106385567675, 19.98802291870321693928896988443, 21.07684633926247556262791874222, 21.94130895355199244545443900882, 22.707914484447246007341244728208, 23.2314341577645155512672083857, 24.93619113610864400184428047197

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