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Properties

Label 2-6e2-36.11-c3-0-10
Degree $2$
Conductor $36$
Sign $0.771 + 0.636i$
Analytic cond. $2.12406$
Root an. cond. $1.45741$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.44 − 2.43i)2-s + (1.42 + 4.99i)3-s + (−3.82 − 7.02i)4-s + (14.6 − 8.45i)5-s + (14.2 + 3.74i)6-s + (3.08 + 1.78i)7-s + (−22.6 − 0.866i)8-s + (−22.9 + 14.2i)9-s + (0.610 − 47.8i)10-s + (−25.0 + 43.4i)11-s + (29.6 − 29.1i)12-s + (−18.9 − 32.9i)13-s + (8.80 − 4.93i)14-s + (63.2 + 61.0i)15-s + (−34.7 + 53.7i)16-s + 84.3i·17-s + ⋯
L(s)  = 1  + (0.511 − 0.859i)2-s + (0.275 + 0.961i)3-s + (−0.477 − 0.878i)4-s + (1.31 − 0.756i)5-s + (0.967 + 0.254i)6-s + (0.166 + 0.0963i)7-s + (−0.999 − 0.0382i)8-s + (−0.848 + 0.529i)9-s + (0.0193 − 1.51i)10-s + (−0.687 + 1.19i)11-s + (0.713 − 0.701i)12-s + (−0.405 − 0.701i)13-s + (0.168 − 0.0941i)14-s + (1.08 + 1.05i)15-s + (−0.543 + 0.839i)16-s + 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.771 + 0.636i$
Analytic conductor: \(2.12406\)
Root analytic conductor: \(1.45741\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3/2),\ 0.771 + 0.636i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68863 - 0.606565i\)
\(L(\frac12)\) \(\approx\) \(1.68863 - 0.606565i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.44 + 2.43i)T \)
3 \( 1 + (-1.42 - 4.99i)T \)
good5 \( 1 + (-14.6 + 8.45i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-3.08 - 1.78i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (25.0 - 43.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (18.9 + 32.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 84.3iT - 4.91e3T^{2} \)
19 \( 1 + 62.9iT - 6.85e3T^{2} \)
23 \( 1 + (37.6 + 65.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-105. - 60.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-17.2 + 9.97i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 17.7T + 5.06e4T^{2} \)
41 \( 1 + (-299. + 172. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (113. + 65.3i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-153. + 265. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 479. iT - 1.48e5T^{2} \)
59 \( 1 + (-245. - 425. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (49.9 - 86.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-536. + 309. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 254.T + 3.57e5T^{2} \)
73 \( 1 - 100.T + 3.89e5T^{2} \)
79 \( 1 + (856. + 494. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (251. - 436. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.01e3iT - 7.04e5T^{2} \)
97 \( 1 + (503. - 872. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.50660512305298785161308113217, −14.55535994332236557661971622206, −13.33551371043871460363022995872, −12.43026123279801494842590943475, −10.53282825309960958537394220639, −9.904702243897200747513057649298, −8.725343400836210310542714396475, −5.62191211045875069058245901426, −4.60812526214984099353279702918, −2.32654633718595309903669187281, 2.76901326243391386402702945303, 5.63851821535940970446900166573, 6.66173029063509044049575100553, 7.976915233245263823652151063252, 9.515927289510443179515208655969, 11.50168765555277714509884492693, 13.04417187289954909270470029660, 14.02520137457649701341116576964, 14.29973315336724252528064430251, 16.11392732554925705971800937787

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