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Properties

Label 2-39e2-169.18-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 + (0.814 − 0.638i)7-s − i·13-s + (0.354 + 0.935i)16-s + (−0.420 + 0.420i)19-s + (0.663 − 0.748i)25-s + (1.03 − 0.0624i)28-s + (0.0217 − 0.359i)31-s + (−1.57 − 0.0950i)37-s + (−1.31 + 1.48i)43-s + (0.0168 − 0.0685i)49-s + (0.568 − 0.822i)52-s + (0.198 + 1.63i)61-s + (−0.239 + 0.970i)64-s + (1.96 + 0.359i)67-s + ⋯
L(s)  = 1  + (0.822 + 0.568i)4-s + (0.814 − 0.638i)7-s − i·13-s + (0.354 + 0.935i)16-s + (−0.420 + 0.420i)19-s + (0.663 − 0.748i)25-s + (1.03 − 0.0624i)28-s + (0.0217 − 0.359i)31-s + (−1.57 − 0.0950i)37-s + (−1.31 + 1.48i)43-s + (0.0168 − 0.0685i)49-s + (0.568 − 0.822i)52-s + (0.198 + 1.63i)61-s + (−0.239 + 0.970i)64-s + (1.96 + 0.359i)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: $\chi_{1521} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.998 - 0.0557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.438270884\)
\(L(\frac12)\) \(\approx\) \(1.438270884\)
\(L(1)\) 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 \( 1 + (-0.814 + 0.638i)T + (0.239 - 0.970i)T^{2} \)
11 \( 1 + (-0.822 + 0.568i)T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 + (0.420 - 0.420i)T - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.568 - 0.822i)T^{2} \)
31 \( 1 + (-0.0217 + 0.359i)T + (-0.992 - 0.120i)T^{2} \)
37 \( 1 + (1.57 + 0.0950i)T + (0.992 + 0.120i)T^{2} \)
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 \( 1 + (-1.96 - 0.359i)T + (0.935 + 0.354i)T^{2} \)
71 \( 1 + (-0.464 - 0.885i)T^{2} \)
73 \( 1 + (0.244 + 0.783i)T + (-0.822 + 0.568i)T^{2} \)
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 \( 1 + (1.28 + 0.580i)T + (0.663 + 0.748i)T^{2} \)
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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

−9.871638569886952888998471167068, −8.453494457781519409479525419617, −8.153102145376438863516655432176, −7.29537267410279956553151802808, −6.59193486590815149222285789764, −5.60158821024763712109143030188, −4.57967361139642501902880312566, −3.61365447263377001311814197303, −2.64409156811188820116871328243, −1.44699063334367788913409189890, 1.59110588183866676343589407103, 2.31073351324051908347344437371, 3.58652353903822786355461793421, 4.97231967631937263609773395572, 5.37706278458280935347316304340, 6.64151988814271269201264897586, 6.95010524823640542985662422775, 8.133772458348117096298844535513, 8.849530022220242526684691926738, 9.635029379269668247028347388892

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