Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.855 + 0.517i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.96 + 0.525i)2-s + (−4.76 + 7.63i)3-s + (15.4 − 4.17i)4-s + (11.0 + 19.1i)5-s + (14.8 − 32.7i)6-s + (−82.7 − 47.7i)7-s + (−59.0 + 24.6i)8-s + (−35.5 − 72.7i)9-s + (−54.0 − 70.2i)10-s + (18.9 + 10.9i)11-s + (−41.8 + 137. i)12-s + (−63.1 − 109. i)13-s + (353. + 146. i)14-s + (−199. − 6.89i)15-s + (221. − 128. i)16-s − 283.·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.131i)2-s + (−0.529 + 0.848i)3-s + (0.965 − 0.260i)4-s + (0.442 + 0.767i)5-s + (0.413 − 0.910i)6-s + (−1.68 − 0.975i)7-s + (−0.922 + 0.385i)8-s + (−0.438 − 0.898i)9-s + (−0.540 − 0.702i)10-s + (0.156 + 0.0903i)11-s + (−0.290 + 0.956i)12-s + (−0.373 − 0.646i)13-s + (1.80 + 0.744i)14-s + (−0.885 − 0.0306i)15-s + (0.864 − 0.503i)16-s − 0.982·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.855 + 0.517i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.855 + 0.517i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0243026 - 0.0872145i\)
\(L(\frac12)\)  \(\approx\)  \(0.0243026 - 0.0872145i\)
\(L(3)\)   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 \( 1 + (3.96 - 0.525i)T \)
3 \( 1 + (4.76 - 7.63i)T \)
good5 \( 1 + (-11.0 - 19.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (82.7 + 47.7i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-18.9 - 10.9i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (63.1 + 109. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 283.T + 8.35e4T^{2} \)
19 \( 1 - 323. iT - 1.30e5T^{2} \)
23 \( 1 + (198. - 114. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (604. - 1.04e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (718. - 414. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 318.T + 1.87e6T^{2} \)
41 \( 1 + (164. + 284. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-179. - 103. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.06e3 - 613. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.83e3T + 7.89e6T^{2} \)
59 \( 1 + (1.27e3 - 737. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-936. + 1.62e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (214. - 123. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 4.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.01e3T + 2.83e7T^{2} \)
79 \( 1 + (6.22e3 + 3.59e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.87e3 + 1.66e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 1.54e3T + 6.27e7T^{2} \)
97 \( 1 + (2.91e3 - 5.05e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−16.48297450234152402341651598320, −15.66207983237158688241940457113, −14.37897571165810490447749079479, −12.57331216172531910897126688827, −10.80195361791699416858016495260, −10.20775977541380796340472621240, −9.286509085174083840326886578043, −7.05550227006094234061587663890, −6.07479233378525649103399204896, −3.32542898187643973377216373349, 0.083892274039083929075577699879, 2.29502383397840835641640540547, 5.89959207274497492798228899846, 6.92538841069092399333651234995, 8.805763361493520075578282532355, 9.619783435256549651413619289444, 11.40410694497303556500477409022, 12.50681959504644530178295346374, 13.25190790605855177116343498711, 15.55317312362124818012747189856

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