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Properties

Label 2-39e2-13.5-c0-0-2
Degree $2$
Conductor $1521$
Sign $0.957 + 0.289i$
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  + (−1 + i)2-s − i·4-s + (1 − i)5-s + 2i·10-s + (−1 − i)11-s + 16-s + (−1 − i)20-s + 2·22-s − i·25-s + (−1 + i)32-s + (1 − i)41-s − 2i·43-s + (−1 + i)44-s + (−1 − i)47-s − i·49-s + (1 + i)50-s + ⋯
L(s)  = 1  + (−1 + i)2-s − i·4-s + (1 − i)5-s + 2i·10-s + (−1 − i)11-s + 16-s + (−1 − i)20-s + 2·22-s − i·25-s + (−1 + i)32-s + (1 − i)41-s − 2i·43-s + (−1 + i)44-s + (−1 − i)47-s − i·49-s + (1 + i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6651699422\)
\(L(\frac12)\) \(\approx\) \(0.6651699422\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (1 - i)T - iT^{2} \)
5 \( 1 + (-1 + i)T - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.341202702754915649874651990744, −8.705733363770225578883724166419, −8.252786851167246921159116043252, −7.34551415863903369728697565142, −6.43545905638323584997957937306, −5.53749890009658633194508693005, −5.24258287275329525329409451109, −3.65644961633868683754279358263, −2.21690085741720950086207860267, −0.75235534236804059763170543133, 1.57872489189144797992384118298, 2.50821271977971702998372531733, 3.04860075508357261789170127706, 4.61359621105249802277571192052, 5.71928271138672447523973878824, 6.51889451029998174486108124524, 7.54807602219051054544317013923, 8.192329102445047568091107281418, 9.385502418663857334519916812848, 9.784916867215331326238519325768

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