Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.286 - 0.958i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.42 + 2.05i)2-s + (2.72 + 8.57i)3-s + (7.52 + 14.1i)4-s + (16.6 − 28.7i)5-s + (−8.29 + 35.0i)6-s + (−39.9 + 23.0i)7-s + (−3.25 + 63.9i)8-s + (−66.0 + 46.8i)9-s + (116. − 64.4i)10-s + (63.6 − 36.7i)11-s + (−100. + 103. i)12-s + (151. − 262. i)13-s + (−184. − 3.13i)14-s + (292. + 63.9i)15-s + (−142. + 212. i)16-s − 182.·17-s + ⋯
L(s)  = 1  + (0.857 + 0.514i)2-s + (0.303 + 0.952i)3-s + (0.470 + 0.882i)4-s + (0.664 − 1.15i)5-s + (−0.230 + 0.973i)6-s + (−0.815 + 0.471i)7-s + (−0.0508 + 0.998i)8-s + (−0.816 + 0.578i)9-s + (1.16 − 0.644i)10-s + (0.525 − 0.303i)11-s + (−0.698 + 0.715i)12-s + (0.896 − 1.55i)13-s + (−0.941 − 0.0159i)14-s + (1.29 + 0.284i)15-s + (−0.557 + 0.830i)16-s − 0.629·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.286 - 0.958i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.286 - 0.958i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.97341 + 1.47007i\)
\(L(\frac12)\)  \(\approx\)  \(1.97341 + 1.47007i\)
\(L(3)\)   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 \( 1 + (-3.42 - 2.05i)T \)
3 \( 1 + (-2.72 - 8.57i)T \)
good5 \( 1 + (-16.6 + 28.7i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (39.9 - 23.0i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-63.6 + 36.7i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-151. + 262. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 182.T + 8.35e4T^{2} \)
19 \( 1 + 314. iT - 1.30e5T^{2} \)
23 \( 1 + (-290. - 167. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-357. - 618. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (985. + 568. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.00e3T + 1.87e6T^{2} \)
41 \( 1 + (557. - 965. i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (2.18e3 - 1.25e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (980. - 566. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.05e3T + 7.89e6T^{2} \)
59 \( 1 + (878. + 507. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (430. + 745. i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (559. + 322. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 9.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.89e3T + 2.83e7T^{2} \)
79 \( 1 + (-6.76e3 + 3.90e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-7.05e3 + 4.07e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 7.65e3T + 6.27e7T^{2} \)
97 \( 1 + (6.36e3 + 1.10e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.87345605301220109604944911428, −14.96605682084587813996977017159, −13.41170288697562163670655394757, −12.87239611229409154715933057672, −11.13174911893306988570506243374, −9.355754904797968454013653325072, −8.438214268414614102946714043915, −6.05285653451293899530151625779, −4.96802502831443912816764295549, −3.21150989523953092923165044395, 1.93021883501458398461808737503, 3.57763305076047535447314847388, 6.39023900439188777307986565950, 6.78512347703626091223620255372, 9.370425290633664459956824162952, 10.76054352362897152896857058224, 11.96765438702680631317849783964, 13.34059901557505813398674540699, 13.96637087435849765904887529242, 14.85658879393342998693847891725

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