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Properties

Label 2-1452-33.32-c1-0-10
Degree $2$
Conductor $1452$
Sign $0.418 - 0.907i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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  + (−1.05 + 1.37i)3-s − 0.414i·5-s − 1.74i·7-s + (−0.784 − 2.89i)9-s + 2.25i·13-s + (0.569 + 0.435i)15-s − 6.02·17-s + 3.55i·19-s + (2.40 + 1.84i)21-s + 1.03i·23-s + 4.82·25-s + (4.80 + 1.96i)27-s + 8.46·29-s + 1.08·31-s − 0.724·35-s + ⋯
L(s)  = 1  + (−0.607 + 0.794i)3-s − 0.185i·5-s − 0.660i·7-s + (−0.261 − 0.965i)9-s + 0.625i·13-s + (0.147 + 0.112i)15-s − 1.46·17-s + 0.815i·19-s + (0.524 + 0.401i)21-s + 0.214i·23-s + 0.965·25-s + (0.925 + 0.379i)27-s + 1.57·29-s + 0.194·31-s − 0.122·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.418 - 0.907i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.418 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160634443\)
\(L(\frac12)\) \(\approx\) \(1.160634443\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.05 - 1.37i)T \)
11 \( 1 \)
good5 \( 1 + 0.414iT - 5T^{2} \)
7 \( 1 + 1.74iT - 7T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 + 6.02T + 17T^{2} \)
19 \( 1 - 3.55iT - 19T^{2} \)
23 \( 1 - 1.03iT - 23T^{2} \)
29 \( 1 - 8.46T + 29T^{2} \)
31 \( 1 - 1.08T + 31T^{2} \)
37 \( 1 - 7.64T + 37T^{2} \)
41 \( 1 + 2.74T + 41T^{2} \)
43 \( 1 + 2.87iT - 43T^{2} \)
47 \( 1 - 6.58iT - 47T^{2} \)
53 \( 1 + 6.94iT - 53T^{2} \)
59 \( 1 - 11.0iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 - 16.1iT - 71T^{2} \)
73 \( 1 + 7.18iT - 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 7.11iT - 89T^{2} \)
97 \( 1 + 2.82T + 97T^{2} \)
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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

−9.767962971719907086032658984356, −8.956498295045744144952560499698, −8.255018554052862672882739104907, −6.96754373579683188348002111375, −6.46391956027837290451407375414, −5.42817524896782148290593633434, −4.43989057763634499803865987024, −4.04568743318069378859762059830, −2.69156174697407713068128968750, −1.01763400526740904974901191390, 0.63188934827682347091977902714, 2.17677756500355146416549298576, 2.93466517497433678039860707655, 4.59717240108577219078180136359, 5.21226395498142210035783582905, 6.41943223093875601232684413615, 6.61869870610684780133670102933, 7.76204594366414028015863693767, 8.493922444416172829622785059683, 9.227878146132794962601938875027

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