Properties

Label 2-1332-37.27-c1-0-15
Degree $2$
Conductor $1332$
Sign $-0.367 + 0.929i$
Analytic cond. $10.6360$
Root an. cond. $3.26129$
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  + (3 − 1.73i)5-s + (−0.5 − 0.866i)7-s − 6·11-s + (1.5 − 0.866i)13-s + (−6 − 3.46i)17-s + (3 − 1.73i)19-s − 3.46i·23-s + (3.5 − 6.06i)25-s − 6.92i·29-s − 1.73i·31-s + (−3 − 1.73i)35-s + (−5 − 3.46i)37-s + (6 + 10.3i)41-s + 12.1i·43-s − 12·47-s + ⋯
L(s)  = 1  + (1.34 − 0.774i)5-s + (−0.188 − 0.327i)7-s − 1.80·11-s + (0.416 − 0.240i)13-s + (−1.45 − 0.840i)17-s + (0.688 − 0.397i)19-s − 0.722i·23-s + (0.700 − 1.21i)25-s − 1.28i·29-s − 0.311i·31-s + (−0.507 − 0.292i)35-s + (−0.821 − 0.569i)37-s + (0.937 + 1.62i)41-s + 1.84i·43-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(10.6360\)
Root analytic conductor: \(3.26129\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :1/2),\ -0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.468876456\)
\(L(\frac12)\) \(\approx\) \(1.468876456\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + (5 + 3.46i)T \)
good5 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
41 \( 1 + (-6 - 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9 + 5.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.73iT - 97T^{2} \)
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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

−9.601514581235068858591060583468, −8.577989177628591583292633141466, −7.901773646611723802970746120283, −6.79454192223965092061557930258, −5.95908007854455860808213595867, −5.12968150667544495716855607748, −4.54535514719877011964644927438, −2.88023263124936411377127894985, −2.11775527186064073100278657050, −0.55581628624613320333762480159, 1.84729883422493415750899819092, 2.57951132353270871572217226004, 3.61690768228872792689782712373, 5.18948788014104941735502270143, 5.62721942520925047594118687650, 6.56806223312539988361369026567, 7.26128278000413378372406900213, 8.384362691738263242793166804783, 9.117080672245857976845239242100, 9.989027815712174465727264924053

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