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Properties

Label 2-1452-11.3-c1-0-5
Degree $2$
Conductor $1452$
Sign $0.394 - 0.918i$
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  + (−0.309 − 0.951i)3-s + (3.11 + 2.26i)5-s + (0.0729 − 0.224i)7-s + (−0.809 + 0.587i)9-s + (−5.04 + 3.66i)13-s + (1.19 − 3.66i)15-s + (1.92 + 1.40i)17-s + (1.57 + 4.84i)19-s − 0.236·21-s − 0.236·23-s + (3.04 + 9.37i)25-s + (0.809 + 0.587i)27-s + (−1.38 + 4.25i)29-s + (5.16 − 3.75i)31-s + (0.736 − 0.534i)35-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (1.39 + 1.01i)5-s + (0.0275 − 0.0848i)7-s + (−0.269 + 0.195i)9-s + (−1.39 + 1.01i)13-s + (0.307 − 0.946i)15-s + (0.467 + 0.339i)17-s + (0.360 + 1.11i)19-s − 0.0515·21-s − 0.0492·23-s + (0.609 + 1.87i)25-s + (0.155 + 0.113i)27-s + (−0.256 + 0.789i)29-s + (0.927 − 0.673i)31-s + (0.124 − 0.0903i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.746046335\)
\(L(\frac12)\) \(\approx\) \(1.746046335\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (-3.11 - 2.26i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0729 + 0.224i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.92 - 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.57 - 4.84i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.16 + 3.75i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.16 - 3.57i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.92 + 9.00i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.47T + 43T^{2} \)
47 \( 1 + (-1.11 - 3.44i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.30 - 3.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.663 + 2.04i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.11 + 1.53i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.145T + 67T^{2} \)
71 \( 1 + (-1.11 - 0.812i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1 - 3.07i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.5 + 8.36i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.5 - 9.09i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (-4.11 + 2.99i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.868580927501903139166469285916, −9.041503993109171250938765754953, −7.83783935862698822037031377734, −7.09839736957185946215304911393, −6.44643350773625401568692874658, −5.73972268359706242541728341771, −4.85866180226035427214763168660, −3.40306754220291951483772258890, −2.34037600454252651588676793679, −1.62212526928790415490732355555, 0.69679904787164105524855456559, 2.16660782672772049825811370098, 3.13085039811564968384417726081, 4.79154597316741634561137024958, 5.03895493278672741563355270552, 5.82271407329673163728106334565, 6.80269120575895825346546879184, 7.928195321551980937520598824417, 8.759355119826319804518172888598, 9.659664171699418818740390774607

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