Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.312 + 0.949i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.59 − 1.20i)2-s + (2.60 − 1.49i)3-s + (1.07 + 3.85i)4-s + (−0.890 − 5.05i)5-s + (−5.95 − 0.768i)6-s + (0.0265 − 0.0315i)7-s + (2.94 − 7.43i)8-s + (4.54 − 7.76i)9-s + (−4.68 + 9.12i)10-s + (4.18 + 0.737i)11-s + (8.55 + 8.41i)12-s + (−1.91 − 0.695i)13-s + (−0.0804 + 0.0182i)14-s + (−9.85 − 11.8i)15-s + (−13.6 + 8.29i)16-s + (−15.1 − 26.2i)17-s + ⋯
L(s)  = 1  + (−0.796 − 0.604i)2-s + (0.867 − 0.497i)3-s + (0.269 + 0.963i)4-s + (−0.178 − 1.01i)5-s + (−0.991 − 0.128i)6-s + (0.00378 − 0.00451i)7-s + (0.367 − 0.929i)8-s + (0.504 − 0.863i)9-s + (−0.468 + 0.912i)10-s + (0.380 + 0.0670i)11-s + (0.712 + 0.701i)12-s + (−0.147 − 0.0535i)13-s + (−0.00574 + 0.00130i)14-s + (−0.657 − 0.787i)15-s + (−0.855 + 0.518i)16-s + (−0.890 − 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.312 + 0.949i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.312 + 0.949i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.683402 - 0.944306i\)
\(L(\frac12)\)  \(\approx\)  \(0.683402 - 0.944306i\)
\(L(2)\)   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 \( 1 + (1.59 + 1.20i)T \)
3 \( 1 + (-2.60 + 1.49i)T \)
good5 \( 1 + (0.890 + 5.05i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-0.0265 + 0.0315i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (-4.18 - 0.737i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (1.91 + 0.695i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (15.1 + 26.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (5.91 + 3.41i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-25.3 - 30.2i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (20.7 - 7.53i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-25.1 - 30.0i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-15.9 - 27.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-53.8 - 19.6i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-40.8 - 7.20i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-9.40 + 11.2i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 43.0T + 2.80e3T^{2} \)
59 \( 1 + (40.5 - 7.15i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-14.2 - 11.9i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (-16.7 + 45.9i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (111. - 64.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-63.7 + 110. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-1.11 - 3.05i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (-40.0 - 109. i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (26.9 - 46.6i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (5.60 - 31.7i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.98365652777103443231068250299, −12.16839188899770897147691963899, −11.13317381930098808583991342306, −9.361039136424083504113391082237, −9.105642093621240148864792537216, −7.890940842070475573237747796756, −6.90693591653016054502223449066, −4.51797574765976726569632579924, −2.86684767232365947424284357350, −1.12389187560418900683601165655, 2.39203784285451096293272948117, 4.23710391906247219559516885901, 6.20032394211496047866020917159, 7.29938545841179488201899878822, 8.402809096855132813734203005220, 9.312455261567114493763009942014, 10.54058510130220408844384663987, 11.02719525393203742952019502031, 13.00109280959815052466712811427, 14.37253951606364949398098605077

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