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Properties

Label 2-6e2-36.7-c2-0-3
Degree $2$
Conductor $36$
Sign $0.426 - 0.904i$
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  + (0.523 + 1.93i)2-s + (2.76 + 1.16i)3-s + (−3.45 + 2.02i)4-s + (−4.03 − 6.98i)5-s + (−0.794 + 5.94i)6-s + (3.90 + 2.25i)7-s + (−5.71 − 5.60i)8-s + (6.29 + 6.42i)9-s + (11.3 − 11.4i)10-s + (−3.25 − 1.88i)11-s + (−11.8 + 1.58i)12-s + (−3.52 − 6.10i)13-s + (−2.30 + 8.71i)14-s + (−3.03 − 23.9i)15-s + (7.81 − 13.9i)16-s + 0.517·17-s + ⋯
L(s)  = 1  + (0.261 + 0.965i)2-s + (0.921 + 0.387i)3-s + (−0.862 + 0.505i)4-s + (−0.806 − 1.39i)5-s + (−0.132 + 0.991i)6-s + (0.557 + 0.321i)7-s + (−0.713 − 0.700i)8-s + (0.699 + 0.714i)9-s + (1.13 − 1.14i)10-s + (−0.296 − 0.171i)11-s + (−0.991 + 0.131i)12-s + (−0.271 − 0.469i)13-s + (−0.164 + 0.622i)14-s + (−0.202 − 1.59i)15-s + (0.488 − 0.872i)16-s + 0.0304·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.426 - 0.904i$
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: \((2,\ 36,\ (\ :1),\ 0.426 - 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06770 + 0.677369i\)
\(L(\frac12)\) \(\approx\) \(1.06770 + 0.677369i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.523 - 1.93i)T \)
3 \( 1 + (-2.76 - 1.16i)T \)
good5 \( 1 + (4.03 + 6.98i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.90 - 2.25i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.25 + 1.88i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.52 + 6.10i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 0.517T + 289T^{2} \)
19 \( 1 - 16.4iT - 361T^{2} \)
23 \( 1 + (27.7 - 15.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.48 + 16.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-13.1 + 7.58i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 0.592T + 1.36e3T^{2} \)
41 \( 1 + (-12.3 - 21.4i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (27.8 + 16.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-52.4 - 30.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 0.664T + 2.80e3T^{2} \)
59 \( 1 + (-30.5 + 17.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-33.7 + 58.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-74.4 + 42.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 56.4iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + (126. + 73.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (87.1 + 50.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 25.8T + 7.92e3T^{2} \)
97 \( 1 + (48.2 - 83.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−16.05220697488994088341226002083, −15.49580178715190029392304388528, −14.30845190413975327845878842320, −13.11866659787753243405013608301, −12.04743977622325096029018627346, −9.669338519585950511508174139046, −8.313242553919730451644162653555, −7.915791588035096793756094607338, −5.27230824042880966069573241778, −4.02489187222017173324061173990, 2.59811354296872973494939135639, 4.11394060905594676614654474660, 6.98618152360329267400693972375, 8.344059799917602789161898586456, 10.04480929405673133820795672273, 11.17224269316000093399458752430, 12.31528159877685473232207452990, 13.85200052279850028539300138067, 14.49966445146152986467737063731, 15.46275418538663468838767855953

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