Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.675 - 0.737i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 − 1.12i)3-s + (−2.69 + 3.21i)5-s + (11.1 + 4.05i)7-s + (6.48 + 6.24i)9-s + (7.66 + 9.13i)11-s + (−0.429 + 2.43i)13-s + (11.1 − 5.91i)15-s + (−11.9 − 6.92i)17-s + (1.88 + 3.27i)19-s + (−26.4 − 23.7i)21-s + (9.18 + 25.2i)23-s + (1.28 + 7.30i)25-s + (−11.0 − 24.6i)27-s + (−40.6 + 7.16i)29-s + (32.7 − 11.9i)31-s + ⋯
L(s)  = 1  + (−0.927 − 0.373i)3-s + (−0.539 + 0.642i)5-s + (1.59 + 0.579i)7-s + (0.720 + 0.693i)9-s + (0.696 + 0.830i)11-s + (−0.0330 + 0.187i)13-s + (0.740 − 0.394i)15-s + (−0.705 − 0.407i)17-s + (0.0994 + 0.172i)19-s + (−1.25 − 1.13i)21-s + (0.399 + 1.09i)23-s + (0.0515 + 0.292i)25-s + (−0.409 − 0.912i)27-s + (−1.40 + 0.246i)29-s + (1.05 − 0.385i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.675 - 0.737i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (41, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.675 - 0.737i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.968296 + 0.425959i\)
\(L(\frac12)\)  \(\approx\)  \(0.968296 + 0.425959i\)
\(L(2)\)   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 \( 1 + (2.78 + 1.12i)T \)
good5 \( 1 + (2.69 - 3.21i)T + (-4.34 - 24.6i)T^{2} \)
7 \( 1 + (-11.1 - 4.05i)T + (37.5 + 31.4i)T^{2} \)
11 \( 1 + (-7.66 - 9.13i)T + (-21.0 + 119. i)T^{2} \)
13 \( 1 + (0.429 - 2.43i)T + (-158. - 57.8i)T^{2} \)
17 \( 1 + (11.9 + 6.92i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.88 - 3.27i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.18 - 25.2i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (40.6 - 7.16i)T + (790. - 287. i)T^{2} \)
31 \( 1 + (-32.7 + 11.9i)T + (736. - 617. i)T^{2} \)
37 \( 1 + (-33.5 + 58.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-5.01 - 0.883i)T + (1.57e3 + 574. i)T^{2} \)
43 \( 1 + (35.5 - 29.8i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (-31.9 + 87.6i)T + (-1.69e3 - 1.41e3i)T^{2} \)
53 \( 1 - 51.4iT - 2.80e3T^{2} \)
59 \( 1 + (-9.45 + 11.2i)T + (-604. - 3.42e3i)T^{2} \)
61 \( 1 + (48.1 + 17.5i)T + (2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (-2.35 + 13.3i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (27.9 + 16.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (20.3 + 35.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-12.9 - 73.1i)T + (-5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (-150. + 26.6i)T + (6.47e3 - 2.35e3i)T^{2} \)
89 \( 1 + (18.0 - 10.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (36.5 - 30.6i)T + (1.63e3 - 9.26e3i)T^{2} \)
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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

−13.56931712947319416783857720395, −12.18114133651741609029017900589, −11.46235157788556519341006227082, −10.95320860308134168867042430573, −9.333770091878157909643256868031, −7.78358080820880916373056263579, −7.01665436464934060467191758395, −5.51050094147199569490782590016, −4.34422653265698726691807053005, −1.83845446873197523476360659664, 1.00108318410008881433943335566, 4.16844288953317160554964306030, 4.91503628081004268611125768837, 6.41040992710628807211992158138, 7.913116070309378047258020152490, 8.882701619701281422286889201819, 10.49639405503766256846459311477, 11.31080914130726097493116786490, 11.93790323353280147794084394147, 13.22332854985393661320481808621

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