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Properties

Label 2-39e2-169.151-c0-0-0
Degree $2$
Conductor $1521$
Sign $-0.998 + 0.0557i$
Analytic cond. $0.759077$
Root an. cond. $0.871250$
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  + (−0.822 − 0.568i)4-s + (−1.05 − 1.34i)7-s + i·13-s + (0.354 + 0.935i)16-s + (−1.35 − 1.35i)19-s + (−0.663 + 0.748i)25-s + (0.103 + 1.70i)28-s + (−1.96 − 0.118i)31-s + (0.0744 − 1.23i)37-s + (−1.31 + 1.48i)43-s + (−0.461 + 1.87i)49-s + (0.568 − 0.822i)52-s + (−0.198 − 1.63i)61-s + (0.239 − 0.970i)64-s + (−0.0217 + 0.118i)67-s + ⋯
L(s)  = 1  + (−0.822 − 0.568i)4-s + (−1.05 − 1.34i)7-s + i·13-s + (0.354 + 0.935i)16-s + (−1.35 − 1.35i)19-s + (−0.663 + 0.748i)25-s + (0.103 + 1.70i)28-s + (−1.96 − 0.118i)31-s + (0.0744 − 1.23i)37-s + (−1.31 + 1.48i)43-s + (−0.461 + 1.87i)49-s + (0.568 − 0.822i)52-s + (−0.198 − 1.63i)61-s + (0.239 − 0.970i)64-s + (−0.0217 + 0.118i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.998 + 0.0557i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (658, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ -0.998 + 0.0557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2758137167\)
\(L(\frac12)\) \(\approx\) \(0.2758137167\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 + (0.822 + 0.568i)T^{2} \)
5 \( 1 + (0.663 - 0.748i)T^{2} \)
7 \( 1 + (1.05 + 1.34i)T + (-0.239 + 0.970i)T^{2} \)
11 \( 1 + (0.822 - 0.568i)T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 + (1.35 + 1.35i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.568 - 0.822i)T^{2} \)
31 \( 1 + (1.96 + 0.118i)T + (0.992 + 0.120i)T^{2} \)
37 \( 1 + (-0.0744 + 1.23i)T + (-0.992 - 0.120i)T^{2} \)
41 \( 1 + (0.464 + 0.885i)T^{2} \)
43 \( 1 + (1.31 - 1.48i)T + (-0.120 - 0.992i)T^{2} \)
47 \( 1 + (0.935 + 0.354i)T^{2} \)
53 \( 1 + (-0.970 - 0.239i)T^{2} \)
59 \( 1 + (-0.663 + 0.748i)T^{2} \)
61 \( 1 + (0.198 + 1.63i)T + (-0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.0217 - 0.118i)T + (-0.935 - 0.354i)T^{2} \)
71 \( 1 + (0.464 + 0.885i)T^{2} \)
73 \( 1 + (-1.74 + 0.542i)T + (0.822 - 0.568i)T^{2} \)
79 \( 1 + (0.271 + 0.393i)T + (-0.354 + 0.935i)T^{2} \)
83 \( 1 + (-0.464 + 0.885i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.580 + 1.28i)T + (-0.663 - 0.748i)T^{2} \)
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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

−9.422843096946059046522409100977, −8.726961271457058309706045483855, −7.53691683885945241909898877563, −6.77884245136909374010695850348, −6.13621642538437040396125064042, −4.94613437026387437202021000453, −4.15495880266193092151858536943, −3.49677254663089817504614122943, −1.82644047930744343758119575281, −0.21408100957638061197738678879, 2.20609744094007908907482902592, 3.27512047791043501513248655186, 3.95543570853858975227362591485, 5.27268497748855032160990441706, 5.82943479335683768774371919067, 6.74623228464961488896239501854, 7.971450551557034773599713058376, 8.472534755575083963214507075608, 9.165410759266252689676452744115, 9.937027931608220762748999758091

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