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Properties

Label 2-2268-63.4-c1-0-7
Degree $2$
Conductor $2268$
Sign $-0.0859 - 0.996i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  − 0.866·5-s + (−0.0665 + 2.64i)7-s − 3.51·11-s + (0.933 − 1.61i)13-s + (3.25 − 5.64i)17-s + (2.69 + 4.66i)19-s + 8.64·23-s − 4.24·25-s + (−1.75 − 3.04i)29-s + (0.933 + 1.61i)31-s + (0.0576 − 2.29i)35-s + (1.39 + 2.40i)37-s + (−5.19 + 8.99i)41-s + (2.89 + 5.00i)43-s + (−3.08 + 5.33i)47-s + ⋯
L(s)  = 1  − 0.387·5-s + (−0.0251 + 0.999i)7-s − 1.05·11-s + (0.258 − 0.448i)13-s + (0.790 − 1.36i)17-s + (0.617 + 1.06i)19-s + 1.80·23-s − 0.849·25-s + (−0.326 − 0.565i)29-s + (0.167 + 0.290i)31-s + (0.00975 − 0.387i)35-s + (0.228 + 0.395i)37-s + (−0.810 + 1.40i)41-s + (0.440 + 0.763i)43-s + (−0.449 + 0.778i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0859 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0859 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.0859 - 0.996i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.0859 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.198363698\)
\(L(\frac12)\) \(\approx\) \(1.198363698\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.0665 - 2.64i)T \)
good5 \( 1 + 0.866T + 5T^{2} \)
11 \( 1 + 3.51T + 11T^{2} \)
13 \( 1 + (-0.933 + 1.61i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.25 + 5.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.64T + 23T^{2} \)
29 \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.933 - 1.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.39 - 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.19 - 8.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.89 - 5.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.08 - 5.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.80 - 4.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.82 - 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.14 - 8.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.08T + 71T^{2} \)
73 \( 1 + (3.62 - 6.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.83 - 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.43 + 5.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.28 - 5.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.64 - 2.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.351491236625294126125384156431, −8.240675807638386526372316865947, −7.86017757240941156221446124039, −7.03885057045081071535972356146, −5.85326564131597177680632914693, −5.38397698835387049890098692695, −4.55774062002564963002138793691, −3.12303375875826804180996785833, −2.79770734212208197651167622244, −1.21372064478813345024539015140, 0.45285688780812373793170563951, 1.77941403127079059610127080050, 3.17884899542477397601388240637, 3.81401975377687896298570346889, 4.84359263525688651475763727081, 5.53653954819355375899773659367, 6.68678265976823683759381581232, 7.32314165143348286363536419439, 7.939357040871137218217254985340, 8.757608400834511482551089487453

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