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Properties

Label 2-6e2-36.31-c2-0-9
Degree $2$
Conductor $36$
Sign $-0.550 + 0.834i$
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  + (−0.186 − 1.99i)2-s + (−2.67 − 1.36i)3-s + (−3.93 + 0.741i)4-s + (3.07 − 5.32i)5-s + (−2.21 + 5.57i)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 + (11.5 + 3.38i)12-s + (−0.892 + 1.54i)13-s + (−0.683 − 0.963i)14-s + (−15.4 + 10.0i)15-s + (14.8 − 5.83i)16-s − 16.9·17-s + ⋯
L(s)  = 1  + (−0.0931 − 0.995i)2-s + (−0.890 − 0.454i)3-s + (−0.982 + 0.185i)4-s + (0.614 − 1.06i)5-s + (−0.369 + 0.929i)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 + (0.959 + 0.281i)12-s + (−0.0686 + 0.118i)13-s + (−0.0488 − 0.0688i)14-s + (−1.03 + 0.668i)15-s + (0.931 − 0.364i)16-s − 0.995·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.550 + 0.834i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ -0.550 + 0.834i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.376106 - 0.698751i\)
\(L(\frac12)\) \(\approx\) \(0.376106 - 0.698751i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.186 + 1.99i)T \)
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

−16.48925506478059051650231710818, −14.10472851824038155412881172033, −13.10951039128555646815774481682, −12.13223258868461599131472829840, −11.17193767466084480782769456272, −9.690586620811757851382747186520, −8.451124751536607515974989545642, −6.09116502294459664859252697483, −4.52942972555913344607724757861, −1.34574200029128207473128367631, 4.44257723734643790588761038869, 6.21097102802769350623635352270, 6.96388195830025659024548164248, 9.242801974015537155335507527399, 10.23822003123407077513755530126, 11.66235343602414380813449388551, 13.37275463567049572091705567719, 14.71469610247402702298157044071, 15.37072006799393537809743279991, 16.79958707777961694125903000122

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