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Properties

Label 2-6e2-36.31-c2-0-2
Degree $2$
Conductor $36$
Sign $0.226 - 0.974i$
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  + (1.26 + 1.55i)2-s + (−2.32 + 1.89i)3-s + (−0.806 + 3.91i)4-s + (1.35 − 2.34i)5-s + (−5.87 − 1.20i)6-s + (10.0 − 5.79i)7-s + (−7.09 + 3.70i)8-s + (1.78 − 8.82i)9-s + (5.35 − 0.865i)10-s + (−8.54 + 4.93i)11-s + (−5.56 − 10.6i)12-s + (0.296 − 0.513i)13-s + (21.6 + 8.24i)14-s + (1.31 + 8.03i)15-s + (−14.6 − 6.31i)16-s − 8.87·17-s + ⋯
L(s)  = 1  + (0.631 + 0.775i)2-s + (−0.774 + 0.633i)3-s + (−0.201 + 0.979i)4-s + (0.271 − 0.469i)5-s + (−0.979 − 0.200i)6-s + (1.43 − 0.828i)7-s + (−0.886 + 0.462i)8-s + (0.198 − 0.980i)9-s + (0.535 − 0.0865i)10-s + (−0.777 + 0.448i)11-s + (−0.463 − 0.885i)12-s + (0.0227 − 0.0394i)13-s + (1.54 + 0.588i)14-s + (0.0873 + 0.535i)15-s + (−0.918 − 0.394i)16-s − 0.522·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.226 - 0.974i$
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: \((2,\ 36,\ (\ :1),\ 0.226 - 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.931057 + 0.739441i\)
\(L(\frac12)\) \(\approx\) \(0.931057 + 0.739441i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 1.55i)T \)
3 \( 1 + (2.32 - 1.89i)T \)
good5 \( 1 + (-1.35 + 2.34i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-10.0 + 5.79i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.54 - 4.93i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.296 + 0.513i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 8.87T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (18.2 + 10.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-10.1 - 17.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.3 - 8.27i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 40.6T + 1.36e3T^{2} \)
41 \( 1 + (-21.2 + 36.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (32.2 - 18.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-1.57 + 0.907i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 21.1T + 2.80e3T^{2} \)
59 \( 1 + (-76.6 - 44.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-36.4 - 63.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (38.3 + 22.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 + (8.30 - 4.79i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-73.6 + 42.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 64.7T + 7.92e3T^{2} \)
97 \( 1 + (3.59 + 6.22i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−16.43246952458838673932316076105, −15.40090353195648791200411969709, −14.31300867220342725876669927159, −13.05792099092670821645494403610, −11.71898633231762807577968479031, −10.50257757211474832902501552880, −8.639261111891691394155664857519, −7.11358367916330140416831613862, −5.27845070325643748910069031672, −4.43831727439837761211662327228, 2.11262144784266889695372323625, 4.98734772930607317036881483505, 6.12911409288856878668350704346, 8.189684703082490947534612716129, 10.34056118215821159793240682327, 11.34516308295777813594040412314, 12.14587783020586335916946698208, 13.50155504895738736974688498620, 14.48276677302438344280690379026, 15.78093823498506459781076320651

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