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Properties

Label 2-39e2-13.12-c1-0-46
Degree $2$
Conductor $1521$
Sign $-0.832 + 0.554i$
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  − i·2-s + 4-s + i·5-s − 2i·7-s − 3i·8-s + 10-s − 2i·11-s − 2·14-s − 16-s − 7·17-s − 6i·19-s + i·20-s − 2·22-s − 6·23-s + 4·25-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s + 0.447i·5-s − 0.755i·7-s − 1.06i·8-s + 0.316·10-s − 0.603i·11-s − 0.534·14-s − 0.250·16-s − 1.69·17-s − 1.37i·19-s + 0.223i·20-s − 0.426·22-s − 1.25·23-s + 0.800·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.832 + 0.554i$
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: \((2,\ 1521,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532024170\)
\(L(\frac12)\) \(\approx\) \(1.532024170\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 2T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.228776181286645982006838089994, −8.494602370793332922917974930310, −7.25333726366425297980684803423, −6.85388800529815958650975446941, −6.07086546878824947133683665517, −4.70053793297145509188274322382, −3.82868852592147685335232962155, −2.89081336322744422982611162073, −2.01715652076063376273312109464, −0.55111203215314787658022323502, 1.79178813249096849552527210282, 2.56021628755197362769519055496, 4.05968353687517304113582246009, 5.00366168775714292143470087061, 5.85748183945164244449716008706, 6.49942745626158974401964656137, 7.31369343542435591955382439571, 8.312067228181774045852480533780, 8.625023972237513957465045706406, 9.721887559180964113611639948170

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