Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.986 + 0.162i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−40.7 + 70.5i)5-s + (89.6 + 155. i)7-s + (−250. − 433. i)11-s + (275. − 476. i)13-s − 753.·17-s − 2.57e3·19-s + (−1.37e3 + 2.37e3i)23-s + (−1.75e3 − 3.03e3i)25-s + (1.95e3 + 3.38e3i)29-s + (1.55e3 − 2.68e3i)31-s − 1.46e4·35-s − 9.56e3·37-s + (1.11e3 − 1.92e3i)41-s + (−7.14e3 − 1.23e4i)43-s + (−3.23e3 − 5.60e3i)47-s + ⋯
L(s)  = 1  + (−0.728 + 1.26i)5-s + (0.691 + 1.19i)7-s + (−0.623 − 1.08i)11-s + (0.451 − 0.782i)13-s − 0.632·17-s − 1.63·19-s + (−0.541 + 0.937i)23-s + (−0.561 − 0.972i)25-s + (0.431 + 0.747i)29-s + (0.290 − 0.502i)31-s − 2.01·35-s − 1.14·37-s + (0.103 − 0.179i)41-s + (−0.589 − 1.02i)43-s + (−0.213 − 0.370i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.986 + 0.162i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.986 + 0.162i)\)
\(L(3)\)  \(\approx\)  \(0.0391653 - 0.477840i\)
\(L(\frac12)\)  \(\approx\)  \(0.0391653 - 0.477840i\)
\(L(\frac{7}{2})\)   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 \( 1 \)
good5 \( 1 + (40.7 - 70.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-89.6 - 155. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (250. + 433. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-275. + 476. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 753.T + 1.41e6T^{2} \)
19 \( 1 + 2.57e3T + 2.47e6T^{2} \)
23 \( 1 + (1.37e3 - 2.37e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-1.95e3 - 3.38e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.55e3 + 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 9.56e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.11e3 + 1.92e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (7.14e3 + 1.23e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (3.23e3 + 5.60e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 1.36e4T + 4.18e8T^{2} \)
59 \( 1 + (-2.85e3 + 4.94e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-5.89e3 - 1.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.77e3 + 3.06e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 5.84e4T + 1.80e9T^{2} \)
73 \( 1 + 6.01e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.78e4 - 4.81e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.99e4 - 3.46e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 1.03e5T + 5.58e9T^{2} \)
97 \( 1 + (-8.29e4 - 1.43e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.34942694958684952347569104291, −12.05272492216656238752052399645, −11.10527124744182043769920557225, −10.54043107070156691204560684789, −8.667918406252344873707375395358, −8.010806134666265017402272638694, −6.54891683517000395734192371596, −5.40400664966058854156691841373, −3.57100555346796175631603396324, −2.35762465786687510262631937692, 0.17538671459731487170636270993, 1.72694622891806187122940312168, 4.36090043564385087754783929566, 4.56509140293807029242807548552, 6.69655423375980320536672773163, 7.953099714411364488140315969886, 8.676880255467340734518378087139, 10.16746657179791044506689991568, 11.17592176534151849076951945328, 12.32975506003830386416615857346

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