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Properties

Label 2-39e2-13.12-c1-0-49
Degree $2$
Conductor $1521$
Sign $0.246 + 0.969i$
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  + 1.24i·2-s + 0.445·4-s − 2.80i·5-s − 4.80i·7-s + 3.04i·8-s + 3.49·10-s − 1.46i·11-s + 5.98·14-s − 2.91·16-s − 2.44·17-s − 2.54i·19-s − 1.24i·20-s + 1.82·22-s − 3.51·23-s − 2.85·25-s + ⋯
L(s)  = 1  + 0.881i·2-s + 0.222·4-s − 1.25i·5-s − 1.81i·7-s + 1.07i·8-s + 1.10·10-s − 0.442i·11-s + 1.60·14-s − 0.727·16-s − 0.593·17-s − 0.583i·19-s − 0.278i·20-s + 0.389·22-s − 0.733·23-s − 0.570·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.246 + 0.969i$
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: \((2,\ 1521,\ (\ :1/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.507687885\)
\(L(\frac12)\) \(\approx\) \(1.507687885\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.24iT - 2T^{2} \)
5 \( 1 + 2.80iT - 5T^{2} \)
7 \( 1 + 4.80iT - 7T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + 2.54iT - 19T^{2} \)
23 \( 1 + 3.51T + 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
31 \( 1 - 7.63iT - 31T^{2} \)
37 \( 1 + 4.55iT - 37T^{2} \)
41 \( 1 - 1.24iT - 41T^{2} \)
43 \( 1 + 2.38T + 43T^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 - 8.85T + 53T^{2} \)
59 \( 1 + 2.17iT - 59T^{2} \)
61 \( 1 + 7.82T + 61T^{2} \)
67 \( 1 - 3.58iT - 67T^{2} \)
71 \( 1 + 8.83iT - 71T^{2} \)
73 \( 1 + 7.69iT - 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 + 0.652iT - 83T^{2} \)
89 \( 1 + 6.29iT - 89T^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−8.935190039029009553399711269668, −8.445826363375561259225081080454, −7.53239371823085116744220227449, −7.01501581520938586211024964235, −6.15761090621441656912731349738, −5.12149639383399196336535069998, −4.50324508693855688793303066988, −3.48614795745798756326982555129, −1.81570390387088858324119015285, −0.55143173713508743492152153676, 1.90284245889450972983640169342, 2.50444595797527105925697697939, 3.21698584665955811969621992478, 4.34675030988983904234750974082, 5.78220768140538646343933268262, 6.27014129052476576397118322594, 7.14968798276087998415942439803, 8.067425109994585822082266084438, 9.111572552047070387967444390675, 9.807256480158876835554350602049

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