Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.297 - 0.954i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.40 − 2.75i)3-s + (8.18 + 6.86i)5-s + (−24.4 − 8.90i)7-s + (11.8 + 24.2i)9-s + (0.555 − 0.466i)11-s + (−12.2 + 69.6i)13-s + (−17.1 − 52.8i)15-s + (−42.6 + 73.8i)17-s + (75.7 + 131. i)19-s + (83.2 + 106. i)21-s + (−156. + 57.1i)23-s + (−1.88 − 10.6i)25-s + (14.7 − 139. i)27-s + (−42.3 − 239. i)29-s + (57.5 − 20.9i)31-s + ⋯
L(s)  = 1  + (−0.847 − 0.530i)3-s + (0.732 + 0.614i)5-s + (−1.32 − 0.480i)7-s + (0.437 + 0.899i)9-s + (0.0152 − 0.0127i)11-s + (−0.261 + 1.48i)13-s + (−0.295 − 0.908i)15-s + (−0.607 + 1.05i)17-s + (0.914 + 1.58i)19-s + (0.865 + 1.10i)21-s + (−1.42 + 0.517i)23-s + (−0.0150 − 0.0854i)25-s + (0.105 − 0.994i)27-s + (−0.270 − 1.53i)29-s + (0.333 − 0.121i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.297 - 0.954i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.297 - 0.954i)\)
\(L(2)\)  \(\approx\)  \(0.392625 + 0.533311i\)
\(L(\frac12)\)  \(\approx\)  \(0.392625 + 0.533311i\)
\(L(\frac{5}{2})\)   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 \( 1 + (4.40 + 2.75i)T \)
good5 \( 1 + (-8.18 - 6.86i)T + (21.7 + 123. i)T^{2} \)
7 \( 1 + (24.4 + 8.90i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-0.555 + 0.466i)T + (231. - 1.31e3i)T^{2} \)
13 \( 1 + (12.2 - 69.6i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (42.6 - 73.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-75.7 - 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (156. - 57.1i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (42.3 + 239. i)T + (-2.29e4 + 8.34e3i)T^{2} \)
31 \( 1 + (-57.5 + 20.9i)T + (2.28e4 - 1.91e4i)T^{2} \)
37 \( 1 + (-7.30 + 12.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (13.2 - 75.0i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 + (152. - 128. i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-252. - 91.8i)T + (7.95e4 + 6.67e4i)T^{2} \)
53 \( 1 + 52.6T + 1.48e5T^{2} \)
59 \( 1 + (575. + 482. i)T + (3.56e4 + 2.02e5i)T^{2} \)
61 \( 1 + (34.9 + 12.7i)T + (1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (88.9 - 504. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (-298. + 516. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-187. - 325. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-61.3 - 347. i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (-14.6 - 83.2i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (351. + 608. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-10.3 + 8.70i)T + (1.58e5 - 8.98e5i)T^{2} \)
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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

−13.57592790178018515610518851343, −12.46740371919454817333231268951, −11.53771670617069481040733462913, −10.19764960393452638317921271744, −9.715133776437871127985009385207, −7.72644834612549084931332171427, −6.41445263135171966513450204561, −6.08297203314131292880479569678, −4.00123029995219479885966979579, −1.94527038650278856314605825408, 0.38801109151551445171279231570, 3.03328753714925385929784317049, 4.98113996385288113250967960258, 5.80229972149811916414539826691, 6.99955571284789470772853805029, 9.033517067755091470371243211266, 9.664783973820197406021018480192, 10.64105223166378839660276911856, 12.01767451006454963517175452577, 12.80264025491437049546045075556

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