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Properties

Label 1-363-363.164-r0-0-0
Degree $1$
Conductor $363$
Sign $-0.985 - 0.167i$
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  + (−0.142 − 0.989i)2-s + (−0.959 + 0.281i)4-s + (−0.415 + 0.909i)5-s + (−0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (0.959 − 0.281i)13-s + (−0.415 + 0.909i)14-s + (0.841 − 0.540i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.142 − 0.989i)20-s + (−0.841 + 0.540i)23-s + (−0.654 − 0.755i)25-s + (−0.415 − 0.909i)26-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (−0.959 + 0.281i)4-s + (−0.415 + 0.909i)5-s + (−0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (0.959 + 0.281i)10-s + (0.959 − 0.281i)13-s + (−0.415 + 0.909i)14-s + (0.841 − 0.540i)16-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.142 − 0.989i)20-s + (−0.841 + 0.540i)23-s + (−0.654 − 0.755i)25-s + (−0.415 − 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.985 - 0.167i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ -0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03791924631 - 0.4483109580i\)
\(L(\frac12)\) \(\approx\) \(0.03791924631 - 0.4483109580i\)
\(L(1)\) \(\approx\) \(0.5643214587 - 0.3225278476i\)
\(L(1)\) \(\approx\) \(0.5643214587 - 0.3225278476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.142 - 0.989i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−25.03460025635213919462802369241, −24.33693698913999200930023413828, −23.54129787615493540172305851844, −22.7007889326026990576209857911, −21.87848987411949973054690359538, −20.6887986618555977299927992917, −19.65469565057702937751617770538, −18.82543296821110080807404872500, −17.99075965693034060191436079, −16.84131360769022755962129050996, −16.12838145076227431180769452912, −15.65131307746803790763165880646, −14.566808256708732007349190783220, −13.35683114761876866544759679240, −12.79465560318178101327904134179, −11.7210462781796652497624014198, −10.22651782929959540008504462638, −9.190506824580528653795385752894, −8.565300714044935321241671606831, −7.644071954196735423527810707592, −6.310501985804295374438164723046, −5.71424631771491265802101501577, −4.42280529399229177645780066579, −3.54803661254338876077809646414, −1.48007674254815786686505807772, 0.29879983585409537082463510569, 2.05686864456126815699811317218, 3.378210201627390721328820682060, 3.74178705623379772202025104681, 5.29599629160229568257494547571, 6.72229287536566489903282023406, 7.606784036777739771639400672561, 8.898784724840957772390405363352, 9.80866217164039967853620397489, 10.77645900515064646574062488225, 11.326906619341845937794090323336, 12.4293749168414968105956316512, 13.52486780792303163581403159550, 13.995071707278830493369811548077, 15.42190512741282513350498349388, 16.23798882568736251093483026934, 17.55771672505188248736620708917, 18.31944233523694724740520188894, 19.027034762695024016187760232753, 20.03201033079147586953754053889, 20.419136984100235149310100265142, 21.905847146597263878054614820858, 22.3550560579466832796248709585, 23.139318323235038037764761385036, 23.930881223005958455595039496805

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