Properties

Label 2-39e2-169.86-c0-0-0
Degree $2$
Conductor $1521$
Sign $-0.284 - 0.958i$
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  + (−0.239 + 0.970i)4-s + (0.819 + 1.82i)7-s + i·13-s + (−0.885 − 0.464i)16-s + (−1.11 − 1.11i)19-s + (−0.822 − 0.568i)25-s + (−1.96 + 0.359i)28-s + (0.186 − 1.01i)31-s + (0.807 + 0.147i)37-s + (1.53 + 1.06i)43-s + (−1.97 + 2.23i)49-s + (−0.970 − 0.239i)52-s + (0.169 + 0.447i)61-s + (0.663 − 0.748i)64-s + (1.68 + 1.01i)67-s + ⋯
L(s)  = 1  + (−0.239 + 0.970i)4-s + (0.819 + 1.82i)7-s + i·13-s + (−0.885 − 0.464i)16-s + (−1.11 − 1.11i)19-s + (−0.822 − 0.568i)25-s + (−1.96 + 0.359i)28-s + (0.186 − 1.01i)31-s + (0.807 + 0.147i)37-s + (1.53 + 1.06i)43-s + (−1.97 + 2.23i)49-s + (−0.970 − 0.239i)52-s + (0.169 + 0.447i)61-s + (0.663 − 0.748i)64-s + (1.68 + 1.01i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.284 - 0.958i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ -0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.036485767\)
\(L(\frac12)\) \(\approx\) \(1.036485767\)
\(L(1)\) 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 \( 1 + (0.239 - 0.970i)T^{2} \)
5 \( 1 + (0.822 + 0.568i)T^{2} \)
7 \( 1 + (-0.819 - 1.82i)T + (-0.663 + 0.748i)T^{2} \)
11 \( 1 + (0.239 + 0.970i)T^{2} \)
17 \( 1 + (0.748 + 0.663i)T^{2} \)
19 \( 1 + (1.11 + 1.11i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.970 - 0.239i)T^{2} \)
31 \( 1 + (-0.186 + 1.01i)T + (-0.935 - 0.354i)T^{2} \)
37 \( 1 + (-0.807 - 0.147i)T + (0.935 + 0.354i)T^{2} \)
41 \( 1 + (0.992 + 0.120i)T^{2} \)
43 \( 1 + (-1.53 - 1.06i)T + (0.354 + 0.935i)T^{2} \)
47 \( 1 + (-0.464 - 0.885i)T^{2} \)
53 \( 1 + (-0.748 - 0.663i)T^{2} \)
59 \( 1 + (-0.822 - 0.568i)T^{2} \)
61 \( 1 + (-0.169 - 0.447i)T + (-0.748 + 0.663i)T^{2} \)
67 \( 1 + (-1.68 - 1.01i)T + (0.464 + 0.885i)T^{2} \)
71 \( 1 + (0.992 + 0.120i)T^{2} \)
73 \( 1 + (1.50 + 1.17i)T + (0.239 + 0.970i)T^{2} \)
79 \( 1 + (-1.28 + 0.317i)T + (0.885 - 0.464i)T^{2} \)
83 \( 1 + (-0.992 + 0.120i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.420 - 1.35i)T + (-0.822 + 0.568i)T^{2} \)
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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

−9.416571833774114790261859214071, −9.107835818937661724347602288434, −8.312121936902710123535124743710, −7.78155106644209110986011236776, −6.61413777004465610796918176663, −5.84118785994550063574821194985, −4.72735768435643746744280105515, −4.16820956018082665465489420054, −2.65248830099296009047559570921, −2.16402069706388885519862439355, 0.862742069448653384317684905538, 1.92361089305999699726085057893, 3.68006184048888900606337614167, 4.36334491589433122769255142002, 5.25278452429318894184277176326, 6.09602418645365132031122872382, 7.06745658456971536662610556955, 7.82360789286027383820261564758, 8.515163461678512988620272377845, 9.689024552448337990382028341534

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