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Properties

Label 2-6e2-4.3-c22-0-43
Degree $2$
Conductor $36$
Sign $-0.642 + 0.766i$
Analytic cond. $110.414$
Root an. cond. $10.5078$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85e3 − 865. i)2-s + (2.69e6 − 3.21e6i)4-s − 1.40e7·5-s + 7.81e8i·7-s + (2.21e9 − 8.29e9i)8-s + (−2.61e10 + 1.21e10i)10-s + 4.54e11i·11-s − 1.51e12·13-s + (6.76e11 + 1.45e12i)14-s + (−3.07e12 − 1.73e13i)16-s + 3.34e13·17-s − 1.16e14i·19-s + (−3.79e13 + 4.52e13i)20-s + (3.93e14 + 8.43e14i)22-s − 1.02e15i·23-s + ⋯
L(s)  = 1  + (0.906 − 0.422i)2-s + (0.642 − 0.766i)4-s − 0.288·5-s + 0.395i·7-s + (0.258 − 0.966i)8-s + (−0.261 + 0.121i)10-s + 1.59i·11-s − 0.845·13-s + (0.167 + 0.358i)14-s + (−0.174 − 0.984i)16-s + 0.975·17-s − 1.00i·19-s + (−0.185 + 0.221i)20-s + (0.673 + 1.44i)22-s − 1.07i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(110.414\)
Root analytic conductor: \(10.5078\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :11),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(2.602325627\)
\(L(\frac12)\) \(\approx\) \(2.602325627\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.85e3 + 865. i)T \)
3 \( 1 \)
good5 \( 1 + 1.40e7T + 2.38e15T^{2} \)
7 \( 1 - 7.81e8iT - 3.90e18T^{2} \)
11 \( 1 - 4.54e11iT - 8.14e22T^{2} \)
13 \( 1 + 1.51e12T + 3.21e24T^{2} \)
17 \( 1 - 3.34e13T + 1.17e27T^{2} \)
19 \( 1 + 1.16e14iT - 1.35e28T^{2} \)
23 \( 1 + 1.02e15iT - 9.07e29T^{2} \)
29 \( 1 - 2.04e16T + 1.48e32T^{2} \)
31 \( 1 + 3.11e16iT - 6.45e32T^{2} \)
37 \( 1 + 7.54e15T + 3.16e34T^{2} \)
41 \( 1 + 5.27e17T + 3.02e35T^{2} \)
43 \( 1 + 4.97e17iT - 8.63e35T^{2} \)
47 \( 1 + 1.07e18iT - 6.11e36T^{2} \)
53 \( 1 + 5.00e18T + 8.59e37T^{2} \)
59 \( 1 - 1.63e19iT - 9.09e38T^{2} \)
61 \( 1 + 6.52e19T + 1.89e39T^{2} \)
67 \( 1 + 1.63e20iT - 1.49e40T^{2} \)
71 \( 1 + 3.91e20iT - 5.34e40T^{2} \)
73 \( 1 - 2.53e20T + 9.84e40T^{2} \)
79 \( 1 + 4.85e20iT - 5.59e41T^{2} \)
83 \( 1 - 1.58e21iT - 1.65e42T^{2} \)
89 \( 1 - 3.37e20T + 7.70e42T^{2} \)
97 \( 1 - 4.16e20T + 5.11e43T^{2} \)
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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

−12.00465111744293435552343934781, −10.43533891171924744830648824683, −9.517371949825066058729068829196, −7.64704172334001964199757005216, −6.55261528380248508947338285527, −5.09284995946400482897237269709, −4.30571223126454987932139789835, −2.80654248949272756594342560485, −1.91495291392786201364572966395, −0.38456445254406080674523424799, 1.24460199533753898200705435933, 2.96832720983697464615237661880, 3.76963405971642755220313558816, 5.16135974284923372528643988410, 6.16368062041010677737974352927, 7.50100688955575855230853361977, 8.382895249633662000796387849195, 10.21912210766330775334527466349, 11.52348601152684201456064276254, 12.37112798716303979258740176556

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