Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.670 + 0.742i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.58 − 3.04i)2-s + (−2.22 − 8.72i)3-s + (−2.59 − 15.7i)4-s + (−5.51 + 9.55i)5-s + (−32.3 − 15.8i)6-s + (10.3 − 5.95i)7-s + (−54.8 − 32.9i)8-s + (−71.1 + 38.7i)9-s + (14.8 + 41.5i)10-s + (189. − 109. i)11-s + (−131. + 57.7i)12-s + (18.5 − 32.1i)13-s + (8.54 − 46.8i)14-s + (95.5 + 26.8i)15-s + (−242. + 82.0i)16-s + 284.·17-s + ⋯
L(s)  = 1  + (0.647 − 0.762i)2-s + (−0.246 − 0.969i)3-s + (−0.162 − 0.986i)4-s + (−0.220 + 0.382i)5-s + (−0.898 − 0.439i)6-s + (0.210 − 0.121i)7-s + (−0.857 − 0.514i)8-s + (−0.878 + 0.478i)9-s + (0.148 + 0.415i)10-s + (1.57 − 0.906i)11-s + (−0.916 + 0.400i)12-s + (0.109 − 0.189i)13-s + (0.0435 − 0.239i)14-s + (0.424 + 0.119i)15-s + (−0.947 + 0.320i)16-s + 0.982·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 + 0.742i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.670 + 0.742i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.670 + 0.742i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.703754 - 1.58374i\)
\(L(\frac12)\)  \(\approx\)  \(0.703754 - 1.58374i\)
\(L(3)\)   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 \( 1 + (-2.58 + 3.04i)T \)
3 \( 1 + (2.22 + 8.72i)T \)
good5 \( 1 + (5.51 - 9.55i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-10.3 + 5.95i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-189. + 109. i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-18.5 + 32.1i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 284.T + 8.35e4T^{2} \)
19 \( 1 + 45.4iT - 1.30e5T^{2} \)
23 \( 1 + (-174. - 100. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-614. - 1.06e3i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (1.31e3 + 757. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.52e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.31e3 + 2.28e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (34.6 - 19.9i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.49e3 - 1.44e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 1.41e3T + 7.89e6T^{2} \)
59 \( 1 + (-2.45e3 - 1.41e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.62e3 - 4.55e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-805. - 465. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.16e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.16e3T + 2.83e7T^{2} \)
79 \( 1 + (-6.48e3 + 3.74e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (966. - 558. i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 6.73e3T + 6.27e7T^{2} \)
97 \( 1 + (6.02e3 + 1.04e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−14.64591924103557198824002862427, −14.02193595067474315245564865596, −12.72116625142105726745953684242, −11.67548456576724026823854311639, −10.85333297953509498808537047946, −8.935680008294098699039046765661, −7.00025573010164278166732292611, −5.64175644638566819704428436329, −3.40587114184116586739997339569, −1.21567446135793135053540431719, 3.82216700335430276884032313659, 5.00828264382562702434096131457, 6.59913099543606923645183606326, 8.442242232892192252704762345039, 9.659365500034759210970472617221, 11.59153542260208858067494456311, 12.42466301406409362331267242619, 14.28910876978020736746404421042, 14.87865301507333998871096847956, 16.14320780891447818641996564987

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