Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.986 - 0.163i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 3.31i)2-s + (3.18 − 8.41i)3-s + (−6.00 − 14.8i)4-s + (−23.3 + 40.4i)5-s + (20.7 + 29.3i)6-s + (−52.4 + 30.2i)7-s + (62.6 + 13.2i)8-s + (−60.7 − 53.6i)9-s + (−81.9 − 167. i)10-s + (−63.7 + 36.8i)11-s + (−143. + 3.29i)12-s + (15.5 − 27.0i)13-s + (16.8 − 241. i)14-s + (266. + 325. i)15-s + (−183. + 178. i)16-s + 53.8·17-s + ⋯
L(s)  = 1  + (−0.558 + 0.829i)2-s + (0.353 − 0.935i)3-s + (−0.375 − 0.926i)4-s + (−0.933 + 1.61i)5-s + (0.577 + 0.816i)6-s + (−1.07 + 0.617i)7-s + (0.978 + 0.206i)8-s + (−0.749 − 0.662i)9-s + (−0.819 − 1.67i)10-s + (−0.526 + 0.304i)11-s + (−0.999 + 0.0228i)12-s + (0.0922 − 0.159i)13-s + (0.0857 − 1.23i)14-s + (1.18 + 1.44i)15-s + (−0.718 + 0.695i)16-s + 0.186·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.986 - 0.163i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.986 - 0.163i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0329913 + 0.400566i\)
\(L(\frac12)\)  \(\approx\)  \(0.0329913 + 0.400566i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (2.23 - 3.31i)T \)
3 \( 1 + (-3.18 + 8.41i)T \)
good5 \( 1 + (23.3 - 40.4i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (52.4 - 30.2i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (63.7 - 36.8i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-15.5 + 27.0i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 53.8T + 8.35e4T^{2} \)
19 \( 1 - 54.9iT - 1.30e5T^{2} \)
23 \( 1 + (-243. - 140. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-223. - 387. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-240. - 138. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.01e3T + 1.87e6T^{2} \)
41 \( 1 + (946. - 1.63e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-666. + 384. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.37e3 - 1.37e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 4.64e3T + 7.89e6T^{2} \)
59 \( 1 + (-262. - 151. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (478. + 828. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-6.01e3 - 3.47e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 5.97e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.33e3T + 2.83e7T^{2} \)
79 \( 1 + (3.29e3 - 1.90e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-2.73e3 + 1.57e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 7.13e3T + 6.27e7T^{2} \)
97 \( 1 + (-980. - 1.69e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.98318489920685195583181710137, −15.14064405167663547664941375055, −14.26984655881680122143500209272, −12.82673096456888260193660030959, −11.27117045953503925672716307404, −9.821032904767557345529393335363, −8.161397302273949597702532894244, −7.11475552897643263081937015078, −6.22204458037304513515373829006, −2.97611042074089302971451689249, 0.31381882753901200297430851551, 3.49095042188711499361845057306, 4.69681880143751859151972287356, 7.918967734194729058884038744682, 8.940054660244532766151333434998, 9.940769557227723479183652193685, 11.30668106167387177369003302857, 12.62796415692418297538926315984, 13.52464860269595910360073144851, 15.69774685282361158226378349285

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