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Properties

Label 2-1452-33.32-c1-0-7
Degree $2$
Conductor $1452$
Sign $-0.953 + 0.300i$
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  + (−0.132 + 1.72i)3-s + 2.46i·5-s + 3.58i·7-s + (−2.96 − 0.457i)9-s + 3.89i·13-s + (−4.26 − 0.326i)15-s − 1.15·17-s + 5.29i·19-s + (−6.18 − 0.474i)21-s − 7.11i·23-s − 1.08·25-s + (1.18 − 5.05i)27-s − 2.50·29-s + 5.04·31-s − 8.83·35-s + ⋯
L(s)  = 1  + (−0.0764 + 0.997i)3-s + 1.10i·5-s + 1.35i·7-s + (−0.988 − 0.152i)9-s + 1.07i·13-s + (−1.10 − 0.0844i)15-s − 0.281·17-s + 1.21i·19-s + (−1.34 − 0.103i)21-s − 1.48i·23-s − 0.217·25-s + (0.227 − 0.973i)27-s − 0.465·29-s + 0.906·31-s − 1.49·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.953 + 0.300i$
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: \((2,\ 1452,\ (\ :1/2),\ -0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.272345665\)
\(L(\frac12)\) \(\approx\) \(1.272345665\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.132 - 1.72i)T \)
11 \( 1 \)
good5 \( 1 - 2.46iT - 5T^{2} \)
7 \( 1 - 3.58iT - 7T^{2} \)
13 \( 1 - 3.89iT - 13T^{2} \)
17 \( 1 + 1.15T + 17T^{2} \)
19 \( 1 - 5.29iT - 19T^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 + 2.50T + 29T^{2} \)
31 \( 1 - 5.04T + 31T^{2} \)
37 \( 1 + 0.399T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 1.31iT - 43T^{2} \)
47 \( 1 - 5.96iT - 47T^{2} \)
53 \( 1 - 2.67iT - 53T^{2} \)
59 \( 1 + 12.3iT - 59T^{2} \)
61 \( 1 - 7.95iT - 61T^{2} \)
67 \( 1 + 3.87T + 67T^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 + 5.24iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 3.61T + 83T^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 - 3.08T + 97T^{2} \)
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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

−10.01139129566829844475788112998, −9.135220537799054881046974278252, −8.652756719142250639642442466661, −7.60583226599265823835619705048, −6.29180819786611113201830941421, −6.10060092697771119032437397680, −4.87990361656554472159206825007, −4.02297968522074885335483091207, −2.93124983801183151470860919281, −2.22970464809381599380569477194, 0.55165265208097902971213058969, 1.28850639903241312618782492817, 2.76625102245573147675420318825, 3.97588300735896586284072929177, 4.98838760514561280827508231713, 5.72844265551306614049760591107, 6.84034718525138116133488871068, 7.49572136840810850432547670966, 8.120357406615349776551910450512, 8.954799895175916418685197866332

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