Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.398 - 0.917i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.99 − 0.277i)2-s + (−3.18 + 8.41i)3-s + (15.8 − 2.21i)4-s + (−23.3 + 40.4i)5-s + (−10.3 + 34.4i)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 + (−31.8 + 140. i)12-s + (15.5 − 27.0i)13-s + (200. − 135. i)14-s + (−266. − 325. i)15-s + (246. − 70.2i)16-s + 53.8·17-s + ⋯
L(s)  = 1  + (0.997 − 0.0694i)2-s + (−0.353 + 0.935i)3-s + (0.990 − 0.138i)4-s + (−0.933 + 1.61i)5-s + (−0.288 + 0.957i)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 + (−0.221 + 0.975i)12-s + (0.0922 − 0.159i)13-s + (1.02 − 0.690i)14-s + (−1.18 − 1.44i)15-s + (0.961 − 0.274i)16-s + 0.186·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.398 - 0.917i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.398 - 0.917i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.82146 + 1.19428i\)
\(L(\frac12)\)  \(\approx\)  \(1.82146 + 1.19428i\)
\(L(3)\)   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 \( 1 + (-3.99 + 0.277i)T \)
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

−15.51107478535918917155532714510, −14.68799196276349478789593825026, −14.09925957559109535013571606311, −11.83186068407477823103973266923, −11.12693662163646818461483117769, −10.40468699408531994876671794377, −7.76765204021640225132233007805, −6.36726724767779227500942472725, −4.47088805150040357347699176776, −3.30802580462597037550877908189, 1.53319744853450624304702618224, 4.47259342319050385660810045043, 5.63091288961579056251263783611, 7.56425351631552241147036230949, 8.549342851756299849274346728520, 11.44707944534866003257574673314, 12.02140255715991558118553862271, 12.81325220668577884120895547168, 14.11718604711222131411240690493, 15.43595081202886624838778157466

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