Properties

Label 2-6e2-36.7-c2-0-2
Degree $2$
Conductor $36$
Sign $0.907 - 0.419i$
Analytic cond. $0.980928$
Root an. cond. $0.990418$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 + 1.15i)2-s + (2.67 − 1.36i)3-s + (1.32 − 3.77i)4-s + (3.07 + 5.32i)5-s + (−2.78 + 5.31i)6-s + (−0.511 − 0.295i)7-s + (2.20 + 7.68i)8-s + (5.27 − 7.29i)9-s + (−11.1 − 5.12i)10-s + (−15.1 − 8.72i)11-s + (−1.61 − 11.8i)12-s + (−0.892 − 1.54i)13-s + (1.17 − 0.110i)14-s + (15.4 + 10.0i)15-s + (−12.5 − 9.98i)16-s − 16.9·17-s + ⋯
L(s)  = 1  + (−0.815 + 0.578i)2-s + (0.890 − 0.454i)3-s + (0.330 − 0.943i)4-s + (0.614 + 1.06i)5-s + (−0.463 + 0.886i)6-s + (−0.0730 − 0.0421i)7-s + (0.276 + 0.961i)8-s + (0.586 − 0.810i)9-s + (−1.11 − 0.512i)10-s + (−1.37 − 0.793i)11-s + (−0.134 − 0.990i)12-s + (−0.0686 − 0.118i)13-s + (0.0840 − 0.00785i)14-s + (1.03 + 0.668i)15-s + (−0.781 − 0.624i)16-s − 0.995·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ 0.907 - 0.419i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.930714 + 0.204565i\)
\(L(\frac12)\) \(\approx\) \(0.930714 + 0.204565i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.63 - 1.15i)T \)
3 \( 1 + (-2.67 + 1.36i)T \)
good5 \( 1 + (-3.07 - 5.32i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.511 + 0.295i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (15.1 + 8.72i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.892 + 1.54i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 16.9T + 289T^{2} \)
19 \( 1 - 19.5iT - 361T^{2} \)
23 \( 1 + (-6.86 + 3.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.17 + 5.49i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (27.6 - 15.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 58.2T + 1.36e3T^{2} \)
41 \( 1 + (2.66 + 4.62i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-33.9 - 19.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-9.64 - 5.56i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 35.8T + 2.80e3T^{2} \)
59 \( 1 + (-20.8 + 12.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37.9 - 65.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-31.8 + 18.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 87.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.0T + 5.32e3T^{2} \)
79 \( 1 + (32.1 + 18.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-66.0 - 38.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 27.5T + 7.92e3T^{2} \)
97 \( 1 + (-13.0 + 22.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−16.25172751716111494672437930464, −15.08876058633547078987813408011, −14.20978125527106688332039353996, −13.17646113251987227840619823148, −10.92710551674622477819092656720, −9.928676991465239792193085974513, −8.495979649830652217109152886895, −7.32274525764752044460570805583, −6.03719835621190826860148789830, −2.55650648891392223678608191777, 2.35306030444287242496244825486, 4.69096108301895870049865297326, 7.55056924767514320459927913229, 8.880319984339331217379974398818, 9.616125019791540642781013332349, 10.87806137120566756859799447059, 12.82793412733246489958346771427, 13.32157233555041307695194659963, 15.31500623576593659524529528409, 16.23035015908640897144154056890

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