Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.563 - 0.826i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.42 + 2.05i)2-s + (7.52 − 14.1i)4-s + (−16.6 − 28.7i)5-s + (−39.9 − 23.0i)7-s + (3.25 + 63.9i)8-s + (116. + 64.4i)10-s + (−63.6 − 36.7i)11-s + (151. + 262. i)13-s + (184. − 3.13i)14-s + (−142. − 212. i)16-s + 182.·17-s + 314. i·19-s + (−531. + 18.0i)20-s + (293. − 4.98i)22-s + (−290. + 167. i)23-s + ⋯
L(s)  = 1  + (−0.857 + 0.514i)2-s + (0.470 − 0.882i)4-s + (−0.664 − 1.15i)5-s + (−0.815 − 0.471i)7-s + (0.0508 + 0.998i)8-s + (1.16 + 0.644i)10-s + (−0.525 − 0.303i)11-s + (0.896 + 1.55i)13-s + (0.941 − 0.0159i)14-s + (−0.557 − 0.830i)16-s + 0.629·17-s + 0.870i·19-s + (−1.32 + 0.0450i)20-s + (0.606 − 0.0102i)22-s + (−0.549 + 0.317i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.563 - 0.826i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.563 - 0.826i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.158789 + 0.300457i\)
\(L(\frac12)\)  \(\approx\)  \(0.158789 + 0.300457i\)
\(L(3)\)   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 \( 1 + (3.42 - 2.05i)T \)
3 \( 1 \)
good5 \( 1 + (16.6 + 28.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (39.9 + 23.0i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (63.6 + 36.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-151. - 262. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 182.T + 8.35e4T^{2} \)
19 \( 1 - 314. iT - 1.30e5T^{2} \)
23 \( 1 + (290. - 167. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (357. - 618. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (985. - 568. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.00e3T + 1.87e6T^{2} \)
41 \( 1 + (-557. - 965. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (2.18e3 + 1.25e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-980. - 566. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.05e3T + 7.89e6T^{2} \)
59 \( 1 + (-878. + 507. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (430. - 745. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (559. - 322. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 9.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.89e3T + 2.83e7T^{2} \)
79 \( 1 + (-6.76e3 - 3.90e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (7.05e3 + 4.07e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 7.65e3T + 6.27e7T^{2} \)
97 \( 1 + (6.36e3 - 1.10e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.39177459196467505272135261845, −12.19855563571435629153505920340, −11.12794896712239835708289379020, −9.875649440485436983338059427977, −8.907781904558745326854718525694, −8.033248936996247232548486889043, −6.82391752364714456465660844852, −5.51474418686328614626609043960, −3.90507741931459449007344475971, −1.32342083448177610656987953741, 0.21516999666066585396277696155, 2.68062338186825294117309082483, 3.56011768341787634967059318960, 6.04450498405536966509234251567, 7.33196967771916209383725034687, 8.181366764756818734710191697882, 9.597515841830246478877746645807, 10.54961870392013142219267060753, 11.27756094704435509667415641669, 12.47138997217322359649207167920

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