Properties

Label 2-6e2-36.7-c2-0-8
Degree $2$
Conductor $36$
Sign $-0.173 + 0.984i$
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 − 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (−2 − 3.46i)5-s − 6·6-s + (3 + 1.73i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−3.99 + 6.92i)10-s + (10.5 + 6.06i)11-s + (6.00 + 10.3i)12-s + (11 + 19.0i)13-s − 6.92i·14-s − 12·15-s + (−8 − 13.8i)16-s − 11·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.400 − 0.692i)5-s − 6-s + (0.428 + 0.247i)7-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.399 + 0.692i)10-s + (0.954 + 0.551i)11-s + (0.5 + 0.866i)12-s + (0.846 + 1.46i)13-s − 0.494i·14-s − 0.800·15-s + (−0.5 − 0.866i)16-s − 0.647·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 36,\ (\ :1),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.580462 - 0.691768i\)
\(L(\frac12)\) \(\approx\) \(0.580462 - 0.691768i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
good5 \( 1 + (2 + 3.46i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3 - 1.73i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.5 - 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11 - 19.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 11T + 289T^{2} \)
19 \( 1 + 15.5iT - 361T^{2} \)
23 \( 1 + (21 - 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (17 - 29.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6 - 3.46i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + (6.5 + 11.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-43.5 - 25.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (3 + 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 52T + 2.80e3T^{2} \)
59 \( 1 + (-46.5 + 26.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-8 + 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (100.5 - 58.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + (24 + 13.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (30 + 17.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 2T + 7.92e3T^{2} \)
97 \( 1 + (-21.5 + 37.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−16.18855530086501884641266940690, −14.37964609087059203484598340071, −13.27566539099419178986701490877, −12.10101955844887007024936068437, −11.40119643334500906812118472121, −9.189823897949070557894177217579, −8.581908102134786561740281345875, −7.01049513383616581357229858959, −4.13485203679522370204429728470, −1.70470058177875413975395387501, 3.91804359982382657055054831934, 5.90532203790362723657683539861, 7.74480043160801404547563832259, 8.756213900459044966572019396786, 10.26264197208275269731886275294, 11.13485534000482923608003456392, 13.62032795608416469744064840795, 14.61654249545542069263571442489, 15.37258971442936252342414766868, 16.38441016296190162226586510097

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