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Properties

Label 2-6e2-36.23-c3-0-10
Degree $2$
Conductor $36$
Sign $0.175 + 0.984i$
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  + (−1.65 + 2.29i)2-s + (0.828 − 5.12i)3-s + (−2.52 − 7.58i)4-s + (−14.4 − 8.36i)5-s + (10.4 + 10.3i)6-s + (16.7 − 9.65i)7-s + (21.5 + 6.74i)8-s + (−25.6 − 8.49i)9-s + (43.1 − 19.4i)10-s + (−2.44 − 4.22i)11-s + (−41.0 + 6.68i)12-s + (6.03 − 10.4i)13-s + (−5.50 + 54.3i)14-s + (−54.9 + 67.4i)15-s + (−51.2 + 38.3i)16-s + 71.2i·17-s + ⋯
L(s)  = 1  + (−0.584 + 0.811i)2-s + (0.159 − 0.987i)3-s + (−0.316 − 0.948i)4-s + (−1.29 − 0.748i)5-s + (0.707 + 0.706i)6-s + (0.902 − 0.521i)7-s + (0.954 + 0.298i)8-s + (−0.949 − 0.314i)9-s + (1.36 − 0.613i)10-s + (−0.0669 − 0.115i)11-s + (−0.986 + 0.160i)12-s + (0.128 − 0.223i)13-s + (−0.105 + 1.03i)14-s + (−0.945 + 1.16i)15-s + (−0.800 + 0.599i)16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(2.12406\)
Root analytic conductor: \(1.45741\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3/2),\ 0.175 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.578748 - 0.484862i\)
\(L(\frac12)\) \(\approx\) \(0.578748 - 0.484862i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.65 - 2.29i)T \)
3 \( 1 + (-0.828 + 5.12i)T \)
good5 \( 1 + (14.4 + 8.36i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-16.7 + 9.65i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (2.44 + 4.22i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.03 + 10.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 71.2iT - 4.91e3T^{2} \)
19 \( 1 + 68.3iT - 6.85e3T^{2} \)
23 \( 1 + (-68.0 + 117. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-190. + 109. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-285. - 164. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 133.T + 5.06e4T^{2} \)
41 \( 1 + (29.5 + 17.0i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-0.558 + 0.322i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (93.4 + 161. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 266. iT - 1.48e5T^{2} \)
59 \( 1 + (-104. + 180. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-0.801 - 1.38i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (371. + 214. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 386.T + 3.57e5T^{2} \)
73 \( 1 + 776.T + 3.89e5T^{2} \)
79 \( 1 + (68.5 - 39.5i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-462. - 801. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (-733. - 1.26e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.76472436632209502528355787914, −14.70332716840858932552429539062, −13.48203180661903600438042629214, −12.09077313685066869349803498404, −10.80940575417158669786083669969, −8.480567202403300044281214607559, −8.117318739230695748704305930833, −6.75794022602417947047765373593, −4.71504911010119897971890155827, −0.846166483196409111940889925206, 3.08842216870814418063454795697, 4.57875230324245931592641932825, 7.63862542450230156678737844219, 8.702433048967941365757565394858, 10.17796924634571125285168474040, 11.38814360971250572733357019592, 11.83854087099026060147751029062, 14.04187162993853179681944248010, 15.24652303556199187914821154105, 16.12256849938473417554966615468

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