Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.158 + 0.987i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.84 + 8.13i)3-s + (0.0201 + 0.0239i)5-s + (−60.3 + 21.9i)7-s + (−51.4 − 62.5i)9-s + (122. − 146. i)11-s + (8.21 + 46.5i)13-s + (−0.272 + 0.0716i)15-s + (−383. + 221. i)17-s + (299. − 518. i)19-s + (53.0 − 575. i)21-s + (158. − 435. i)23-s + (108. − 615. i)25-s + (706. − 178. i)27-s + (−396. − 69.9i)29-s + (−1.47e3 − 535. i)31-s + ⋯
L(s)  = 1  + (−0.426 + 0.904i)3-s + (0.000804 + 0.000959i)5-s + (−1.23 + 0.448i)7-s + (−0.635 − 0.771i)9-s + (1.01 − 1.20i)11-s + (0.0486 + 0.275i)13-s + (−0.00121 + 0.000318i)15-s + (−1.32 + 0.766i)17-s + (0.830 − 1.43i)19-s + (0.120 − 1.30i)21-s + (0.299 − 0.823i)23-s + (0.173 − 0.984i)25-s + (0.969 − 0.245i)27-s + (−0.471 − 0.0831i)29-s + (−1.53 − 0.557i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.158 + 0.987i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.158 + 0.987i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.299509 - 0.351477i\)
\(L(\frac12)\)  \(\approx\)  \(0.299509 - 0.351477i\)
\(L(3)\)   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 \( 1 + (3.84 - 8.13i)T \)
good5 \( 1 + (-0.0201 - 0.0239i)T + (-108. + 615. i)T^{2} \)
7 \( 1 + (60.3 - 21.9i)T + (1.83e3 - 1.54e3i)T^{2} \)
11 \( 1 + (-122. + 146. i)T + (-2.54e3 - 1.44e4i)T^{2} \)
13 \( 1 + (-8.21 - 46.5i)T + (-2.68e4 + 9.76e3i)T^{2} \)
17 \( 1 + (383. - 221. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-299. + 518. i)T + (-6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-158. + 435. i)T + (-2.14e5 - 1.79e5i)T^{2} \)
29 \( 1 + (396. + 69.9i)T + (6.64e5 + 2.41e5i)T^{2} \)
31 \( 1 + (1.47e3 + 535. i)T + (7.07e5 + 5.93e5i)T^{2} \)
37 \( 1 + (532. + 922. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (2.32e3 - 410. i)T + (2.65e6 - 9.66e5i)T^{2} \)
43 \( 1 + (-1.34e3 - 1.12e3i)T + (5.93e5 + 3.36e6i)T^{2} \)
47 \( 1 + (-864. - 2.37e3i)T + (-3.73e6 + 3.13e6i)T^{2} \)
53 \( 1 - 2.41e3iT - 7.89e6T^{2} \)
59 \( 1 + (-165. - 197. i)T + (-2.10e6 + 1.19e7i)T^{2} \)
61 \( 1 + (3.68e3 - 1.34e3i)T + (1.06e7 - 8.89e6i)T^{2} \)
67 \( 1 + (702. + 3.98e3i)T + (-1.89e7 + 6.89e6i)T^{2} \)
71 \( 1 + (2.14e3 - 1.23e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-761. + 1.31e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-1.35e3 + 7.66e3i)T + (-3.66e7 - 1.33e7i)T^{2} \)
83 \( 1 + (1.22e4 + 2.15e3i)T + (4.45e7 + 1.62e7i)T^{2} \)
89 \( 1 + (-4.11e3 - 2.37e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (4.42e3 + 3.71e3i)T + (1.53e7 + 8.71e7i)T^{2} \)
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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

−12.62198828392802203867993795861, −11.44925836637989924696568547795, −10.70164257601289949032419691905, −9.203389050801222288347168379692, −8.953105051585970276001631485478, −6.65380485449777242799463478431, −5.91201661442611988480488477305, −4.28129581611387026190494165717, −3.04107956677302507094435404041, −0.21511063684457255220585980475, 1.62689020413903087399978615928, 3.56366817804883437119976670677, 5.40622721578394199263354228555, 6.83125155168638468201717960838, 7.26647319214785869180771257114, 9.049265826840623818382727827012, 10.05319795891679792034696388180, 11.40139876704470028227641021447, 12.34623497295353071321862287070, 13.14101138746379311339232795430

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