Properties

Label 2-6e2-12.11-c15-0-24
Degree $2$
Conductor $36$
Sign $0.770 + 0.637i$
Analytic cond. $51.3696$
Root an. cond. $7.16726$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (179. − 23.8i)2-s + (3.16e4 − 8.54e3i)4-s + 1.77e5i·5-s − 2.64e6i·7-s + (5.47e6 − 2.28e6i)8-s + (4.22e6 + 3.18e7i)10-s + 7.18e7·11-s − 2.80e8·13-s + (−6.30e7 − 4.75e8i)14-s + (9.27e8 − 5.40e8i)16-s − 2.40e9i·17-s + 1.20e9i·19-s + (1.51e9 + 5.61e9i)20-s + (1.28e10 − 1.70e9i)22-s + 2.62e10·23-s + ⋯
L(s)  = 1  + (0.991 − 0.131i)2-s + (0.965 − 0.260i)4-s + 1.01i·5-s − 1.21i·7-s + (0.922 − 0.385i)8-s + (0.133 + 1.00i)10-s + 1.11·11-s − 1.24·13-s + (−0.159 − 1.20i)14-s + (0.864 − 0.503i)16-s − 1.42i·17-s + 0.309i·19-s + (0.264 + 0.980i)20-s + (1.10 − 0.146i)22-s + 1.60·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(51.3696\)
Root analytic conductor: \(7.16726\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :15/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(8)\) \(\approx\) \(4.419387741\)
\(L(\frac12)\) \(\approx\) \(4.419387741\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-179. + 23.8i)T \)
3 \( 1 \)
good5 \( 1 - 1.77e5iT - 3.05e10T^{2} \)
7 \( 1 + 2.64e6iT - 4.74e12T^{2} \)
11 \( 1 - 7.18e7T + 4.17e15T^{2} \)
13 \( 1 + 2.80e8T + 5.11e16T^{2} \)
17 \( 1 + 2.40e9iT - 2.86e18T^{2} \)
19 \( 1 - 1.20e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.62e10T + 2.66e20T^{2} \)
29 \( 1 - 9.72e10iT - 8.62e21T^{2} \)
31 \( 1 + 8.11e10iT - 2.34e22T^{2} \)
37 \( 1 - 7.10e11T + 3.33e23T^{2} \)
41 \( 1 + 9.47e11iT - 1.55e24T^{2} \)
43 \( 1 + 1.52e12iT - 3.17e24T^{2} \)
47 \( 1 + 3.08e12T + 1.20e25T^{2} \)
53 \( 1 + 8.49e12iT - 7.31e25T^{2} \)
59 \( 1 - 5.71e12T + 3.65e26T^{2} \)
61 \( 1 - 4.24e13T + 6.02e26T^{2} \)
67 \( 1 - 6.46e13iT - 2.46e27T^{2} \)
71 \( 1 - 6.28e13T + 5.87e27T^{2} \)
73 \( 1 + 2.15e13T + 8.90e27T^{2} \)
79 \( 1 + 2.44e14iT - 2.91e28T^{2} \)
83 \( 1 + 1.86e14T + 6.11e28T^{2} \)
89 \( 1 - 2.41e14iT - 1.74e29T^{2} \)
97 \( 1 + 5.66e14T + 6.33e29T^{2} \)
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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

−13.18649681455240712513266750344, −11.76558799737060754747635385622, −10.85499353074075385321292848765, −9.724068281119669096058510612879, −7.18048768850187301870971144388, −6.88407972273551314509937729737, −5.00241438065787121388714818748, −3.72433245626517270695780718914, −2.60428097956049615670653764928, −0.909478641567678359375825804253, 1.34229035841720221000810575755, 2.68364715705426324918381882616, 4.34890640538335441596623963030, 5.35081934214490277288307348919, 6.56419435999128357278863435254, 8.252018520750195487892369150381, 9.440731459979787493496372216279, 11.37228489471507050222704072226, 12.37412546987574727805847341442, 12.96921043921606801934524251301

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