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Properties

Label 2-369-9.4-c1-0-9
Degree $2$
Conductor $369$
Sign $0.999 + 0.0252i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.395 − 0.684i)2-s + (−1.70 + 0.279i)3-s + (0.687 − 1.19i)4-s + (−0.0693 + 0.120i)5-s + (0.866 + 1.05i)6-s + (2.32 + 4.02i)7-s − 2.66·8-s + (2.84 − 0.954i)9-s + 0.109·10-s + (−0.684 − 1.18i)11-s + (−0.842 + 2.22i)12-s + (−3.08 + 5.33i)13-s + (1.83 − 3.18i)14-s + (0.0850 − 0.224i)15-s + (−0.321 − 0.556i)16-s + 6.39·17-s + ⋯
L(s)  = 1  + (−0.279 − 0.484i)2-s + (−0.986 + 0.161i)3-s + (0.343 − 0.595i)4-s + (−0.0310 + 0.0537i)5-s + (0.353 + 0.432i)6-s + (0.878 + 1.52i)7-s − 0.943·8-s + (0.948 − 0.318i)9-s + 0.0346·10-s + (−0.206 − 0.357i)11-s + (−0.243 + 0.643i)12-s + (−0.854 + 1.48i)13-s + (0.491 − 0.850i)14-s + (0.0219 − 0.0580i)15-s + (−0.0802 − 0.139i)16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.999 + 0.0252i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.999 + 0.0252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986268 - 0.0124548i\)
\(L(\frac12)\) \(\approx\) \(0.986268 - 0.0124548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.279i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.395 + 0.684i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.0693 - 0.120i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.32 - 4.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.684 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.08 - 5.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.39T + 17T^{2} \)
19 \( 1 - 3.98T + 19T^{2} \)
23 \( 1 + (-2.79 + 4.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.62 - 2.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.67 + 6.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
43 \( 1 + (0.911 + 1.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.89 - 6.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (0.751 - 1.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.283 - 0.490i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.62 + 2.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.51T + 71T^{2} \)
73 \( 1 + 2.79T + 73T^{2} \)
79 \( 1 + (-1.47 - 2.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.39 + 7.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.173T + 89T^{2} \)
97 \( 1 + (-4.90 - 8.48i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.58939777827283803400400683639, −10.66214189652723325528586755774, −9.619796861391115430117152189987, −9.042005737712611233543562062157, −7.58096773040552827039645891916, −6.36011873233499369247159709933, −5.50050964446694268776183657858, −4.80826858211516856628153316739, −2.74207876006419458607058220340, −1.39305856144094617386071280861, 0.979973838076168965257486311897, 3.23147349857186246475388103978, 4.70108437411828508678452760102, 5.58404449524613548911368507523, 6.97003920467333678910625126717, 7.60890157296133683556928983596, 8.021204866697926934345574326282, 9.915105491305756740978488175178, 10.42998005010448271400764999733, 11.46354359755764195721006083268

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