Properties

Label 2-39e2-13.12-c1-0-11
Degree $2$
Conductor $1521$
Sign $-0.691 + 0.722i$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
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  + 1.80i·2-s − 1.24·4-s + 1.44i·5-s + 3.44i·7-s + 1.35i·8-s − 2.60·10-s + 5.18i·11-s − 6.20·14-s − 4.93·16-s − 0.753·17-s − 7.96i·19-s − 1.80i·20-s − 9.34·22-s − 2.82·23-s + 2.91·25-s + ⋯
L(s)  = 1  + 1.27i·2-s − 0.623·4-s + 0.646i·5-s + 1.30i·7-s + 0.479i·8-s − 0.823·10-s + 1.56i·11-s − 1.65·14-s − 1.23·16-s − 0.182·17-s − 1.82i·19-s − 0.402i·20-s − 1.99·22-s − 0.589·23-s + 0.582·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486184839\)
\(L(\frac12)\) \(\approx\) \(1.486184839\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 1.80iT - 2T^{2} \)
5 \( 1 - 1.44iT - 5T^{2} \)
7 \( 1 - 3.44iT - 7T^{2} \)
11 \( 1 - 5.18iT - 11T^{2} \)
17 \( 1 + 0.753T + 17T^{2} \)
19 \( 1 + 7.96iT - 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 3.91T + 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 6.24iT - 37T^{2} \)
41 \( 1 - 1.80iT - 41T^{2} \)
43 \( 1 - 7.09T + 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 + 1.87iT - 59T^{2} \)
61 \( 1 - 3.34T + 61T^{2} \)
67 \( 1 - 4.54iT - 67T^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 - 2.95iT - 73T^{2} \)
79 \( 1 + 9.43T + 79T^{2} \)
83 \( 1 + 6.46iT - 83T^{2} \)
89 \( 1 + 1.15iT - 89T^{2} \)
97 \( 1 + 8.65iT - 97T^{2} \)
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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

−9.719518800965350709827961697482, −8.891516930621033074976266428196, −8.360037282203744467655217899920, −7.17820637206252390613818742613, −6.91579725744595885070731744920, −6.09034477069619678630006253042, −5.11552001072888826541112247010, −4.57736799700435064376539972789, −2.82893593531154431674806716284, −2.15442141077925096724105773819, 0.60450165489209901482015975115, 1.39156330829601708732277765830, 2.77797267102332078703257956137, 3.86436461674460819072309450638, 4.17505597147181952646903374011, 5.59365795538674494871959080542, 6.42807918164682483294406279863, 7.56020525780053532922267981278, 8.299552324280312360657011499180, 9.177245606605690370806338181339

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