Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.120 + 0.992i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 + 0.946i)2-s + (6.20 − 5.04i)4-s + (−2.08 − 1.20i)5-s + (2.30 − 1.32i)7-s + (−11.7 + 19.3i)8-s + (6.70 + 1.23i)10-s + (−24.1 − 41.8i)11-s + (−20.3 + 35.2i)13-s + (−4.88 + 5.72i)14-s + (13.1 − 62.6i)16-s − 36.3i·17-s − 125. i·19-s + (−19.0 + 3.04i)20-s + (103. + 88.6i)22-s + (97.0 − 168. i)23-s + ⋯
L(s)  = 1  + (−0.942 + 0.334i)2-s + (0.776 − 0.630i)4-s + (−0.186 − 0.107i)5-s + (0.124 − 0.0718i)7-s + (−0.520 + 0.853i)8-s + (0.211 + 0.0391i)10-s + (−0.661 − 1.14i)11-s + (−0.434 + 0.752i)13-s + (−0.0931 + 0.109i)14-s + (0.204 − 0.978i)16-s − 0.518i·17-s − 1.51i·19-s + (−0.212 + 0.0340i)20-s + (1.00 + 0.858i)22-s + (0.879 − 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.120 + 0.992i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.120 + 0.992i)\)
\(L(2)\)  \(\approx\)  \(0.409306 - 0.461983i\)
\(L(\frac12)\)  \(\approx\)  \(0.409306 - 0.461983i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.66 - 0.946i)T \)
3 \( 1 \)
good5 \( 1 + (2.08 + 1.20i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-2.30 + 1.32i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (24.1 + 41.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (20.3 - 35.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 36.3iT - 4.91e3T^{2} \)
19 \( 1 + 125. iT - 6.85e3T^{2} \)
23 \( 1 + (-97.0 + 168. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (153. - 88.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (152. + 87.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 199.T + 5.06e4T^{2} \)
41 \( 1 + (-201. - 116. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (252. - 145. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-30.9 - 53.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 352. iT - 1.48e5T^{2} \)
59 \( 1 + (70.7 - 122. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (7.71 + 13.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-131. - 76.0i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 28.0T + 3.57e5T^{2} \)
73 \( 1 - 124.T + 3.89e5T^{2} \)
79 \( 1 + (-648. + 374. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-174. - 302. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 416. iT - 7.04e5T^{2} \)
97 \( 1 + (752. + 1.30e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.94413441725336312758185642316, −11.43195560976885691791672321633, −10.89450231197346073658063573304, −9.493238458761740767740023372790, −8.624037145380679997922952569670, −7.50918882304764694476765267294, −6.37546144874095160795892743829, −4.91599614381794025163951075930, −2.61921667415865357368730733489, −0.44454764896787524085977176106, 1.84721454707034965672713924622, 3.56256464278391803587294740792, 5.56320867444479624526249028763, 7.33409104201783608680772169221, 7.909819200362363599779919957535, 9.399199881853098832791297047549, 10.20903728013452052964985600144, 11.23413011587684417382656196283, 12.34887163062074255009490407184, 13.10692107013749037969674940121

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