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Properties

Label 2-6e2-4.3-c16-0-30
Degree $2$
Conductor $36$
Sign $0.983 + 0.179i$
Analytic cond. $58.4368$
Root an. cond. $7.64439$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (254. + 23.0i)2-s + (6.44e4 + 1.17e4i)4-s + 3.63e5·5-s + 2.07e6i·7-s + (1.61e7 + 4.47e6i)8-s + (9.26e7 + 8.36e6i)10-s − 4.00e8i·11-s + 2.75e8·13-s + (−4.78e7 + 5.29e8i)14-s + (4.01e9 + 1.51e9i)16-s + 7.49e9·17-s − 1.26e10i·19-s + (2.34e10 + 4.26e9i)20-s + (9.22e9 − 1.02e11i)22-s + 1.11e11i·23-s + ⋯
L(s)  = 1  + (0.995 + 0.0898i)2-s + (0.983 + 0.179i)4-s + 0.930·5-s + 0.360i·7-s + (0.963 + 0.266i)8-s + (0.926 + 0.0836i)10-s − 1.87i·11-s + 0.338·13-s + (−0.0323 + 0.358i)14-s + (0.935 + 0.352i)16-s + 1.07·17-s − 0.745i·19-s + (0.915 + 0.166i)20-s + (0.168 − 1.86i)22-s + 1.42i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.983 + 0.179i$
Analytic conductor: \(58.4368\)
Root analytic conductor: \(7.64439\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :8),\ 0.983 + 0.179i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(5.497572886\)
\(L(\frac12)\) \(\approx\) \(5.497572886\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-254. - 23.0i)T \)
3 \( 1 \)
good5 \( 1 - 3.63e5T + 1.52e11T^{2} \)
7 \( 1 - 2.07e6iT - 3.32e13T^{2} \)
11 \( 1 + 4.00e8iT - 4.59e16T^{2} \)
13 \( 1 - 2.75e8T + 6.65e17T^{2} \)
17 \( 1 - 7.49e9T + 4.86e19T^{2} \)
19 \( 1 + 1.26e10iT - 2.88e20T^{2} \)
23 \( 1 - 1.11e11iT - 6.13e21T^{2} \)
29 \( 1 + 4.40e11T + 2.50e23T^{2} \)
31 \( 1 + 9.68e11iT - 7.27e23T^{2} \)
37 \( 1 - 2.64e12T + 1.23e25T^{2} \)
41 \( 1 - 7.96e12T + 6.37e25T^{2} \)
43 \( 1 + 6.27e12iT - 1.36e26T^{2} \)
47 \( 1 - 1.17e13iT - 5.66e26T^{2} \)
53 \( 1 - 5.56e13T + 3.87e27T^{2} \)
59 \( 1 - 2.82e14iT - 2.15e28T^{2} \)
61 \( 1 + 1.86e14T + 3.67e28T^{2} \)
67 \( 1 - 8.81e13iT - 1.64e29T^{2} \)
71 \( 1 + 1.09e15iT - 4.16e29T^{2} \)
73 \( 1 - 5.03e14T + 6.50e29T^{2} \)
79 \( 1 - 1.56e15iT - 2.30e30T^{2} \)
83 \( 1 + 8.07e14iT - 5.07e30T^{2} \)
89 \( 1 - 5.95e15T + 1.54e31T^{2} \)
97 \( 1 + 2.50e15T + 6.14e31T^{2} \)
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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

−13.35429840138457865464162152388, −11.78852985075096755329518569732, −10.82692653205579601108248774421, −9.269658639117848544719570433719, −7.70940416761974139950055531574, −5.94687355953061477745297699307, −5.60502392636508477114750219030, −3.69338002412077385746892148022, −2.55273569661953398138969344949, −1.10239578087477115944774503163, 1.37692140053876578049396255522, 2.37530863622687024323198680737, 3.99151782964427502158868054919, 5.20493144046636271048337618780, 6.41868808969402741712842509127, 7.60643532023464266409353391763, 9.732419896973825125671566842715, 10.53705707471800017342870936452, 12.18653802661867722635919054809, 12.93789355324161537833836869876

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