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Properties

Label 2-39e2-169.8-c0-0-0
Degree $2$
Conductor $1521$
Sign $0.690 - 0.723i$
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  + (−0.992 − 0.120i)4-s + (−0.807 − 0.147i)7-s + i·13-s + (0.970 + 0.239i)16-s + (1.41 + 1.41i)19-s + (−0.464 − 0.885i)25-s + (0.783 + 0.244i)28-s + (0.468 + 1.50i)31-s + (1.87 − 0.585i)37-s + (0.764 + 1.45i)43-s + (−0.305 − 0.115i)49-s + (0.120 − 0.992i)52-s + (−1.12 + 1.63i)61-s + (−0.935 − 0.354i)64-s + (1.17 − 1.50i)67-s + ⋯
L(s)  = 1  + (−0.992 − 0.120i)4-s + (−0.807 − 0.147i)7-s + i·13-s + (0.970 + 0.239i)16-s + (1.41 + 1.41i)19-s + (−0.464 − 0.885i)25-s + (0.783 + 0.244i)28-s + (0.468 + 1.50i)31-s + (1.87 − 0.585i)37-s + (0.764 + 1.45i)43-s + (−0.305 − 0.115i)49-s + (0.120 − 0.992i)52-s + (−1.12 + 1.63i)61-s + (−0.935 − 0.354i)64-s + (1.17 − 1.50i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1360, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7539990470\)
\(L(\frac12)\) \(\approx\) \(0.7539990470\)
\(L(1)\) 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 \( 1 + (0.992 + 0.120i)T^{2} \)
5 \( 1 + (0.464 + 0.885i)T^{2} \)
7 \( 1 + (0.807 + 0.147i)T + (0.935 + 0.354i)T^{2} \)
11 \( 1 + (0.992 - 0.120i)T^{2} \)
17 \( 1 + (0.354 - 0.935i)T^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.120 - 0.992i)T^{2} \)
31 \( 1 + (-0.468 - 1.50i)T + (-0.822 + 0.568i)T^{2} \)
37 \( 1 + (-1.87 + 0.585i)T + (0.822 - 0.568i)T^{2} \)
41 \( 1 + (-0.663 - 0.748i)T^{2} \)
43 \( 1 + (-0.764 - 1.45i)T + (-0.568 + 0.822i)T^{2} \)
47 \( 1 + (0.239 + 0.970i)T^{2} \)
53 \( 1 + (-0.354 + 0.935i)T^{2} \)
59 \( 1 + (-0.464 - 0.885i)T^{2} \)
61 \( 1 + (1.12 - 1.63i)T + (-0.354 - 0.935i)T^{2} \)
67 \( 1 + (-1.17 + 1.50i)T + (-0.239 - 0.970i)T^{2} \)
71 \( 1 + (-0.663 - 0.748i)T^{2} \)
73 \( 1 + (1.70 - 0.103i)T + (0.992 - 0.120i)T^{2} \)
79 \( 1 + (-0.225 - 1.85i)T + (-0.970 + 0.239i)T^{2} \)
83 \( 1 + (0.663 - 0.748i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.731 + 1.21i)T + (-0.464 + 0.885i)T^{2} \)
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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

−9.730326838324816150247758881463, −9.145804257063221306881013401264, −8.211916596872175967909651535250, −7.47544792609596862203850620448, −6.36835986461072645134591532121, −5.73003377164052171967784353680, −4.62880679602714785161882120318, −3.91059868727597390551077714262, −2.95125662623631088038847128563, −1.27891484235901421453083333447, 0.72980728436170777831486963964, 2.74409419765747522474904806798, 3.49401439088465120947116492766, 4.55313743542763898739345633766, 5.42725127659560089276675965048, 6.12395746455242424298767041221, 7.36239337011806535976931449690, 7.912084032336015293613591019676, 8.960058647654543520768885297769, 9.545366240476677097122872720054

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