Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.452 - 0.891i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.96 + 0.525i)2-s + (15.4 + 4.17i)4-s + (−11.0 + 19.1i)5-s + (−82.7 + 47.7i)7-s + (59.0 + 24.6i)8-s + (−54.0 + 70.2i)10-s + (−18.9 + 10.9i)11-s + (−63.1 + 109. i)13-s + (−353. + 146. i)14-s + (221. + 128. i)16-s + 283.·17-s − 323. i·19-s + (−251. + 250. i)20-s + (−80.8 + 33.4i)22-s + (198. + 114. i)23-s + ⋯
L(s)  = 1  + (0.991 + 0.131i)2-s + (0.965 + 0.260i)4-s + (−0.442 + 0.767i)5-s + (−1.68 + 0.975i)7-s + (0.922 + 0.385i)8-s + (−0.540 + 0.702i)10-s + (−0.156 + 0.0903i)11-s + (−0.373 + 0.646i)13-s + (−1.80 + 0.744i)14-s + (0.864 + 0.503i)16-s + 0.982·17-s − 0.896i·19-s + (−0.627 + 0.625i)20-s + (−0.167 + 0.0690i)22-s + (0.375 + 0.216i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.452 - 0.891i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.452 - 0.891i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.13833 + 1.85306i\)
\(L(\frac12)\)  \(\approx\)  \(1.13833 + 1.85306i\)
\(L(3)\)   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 \( 1 + (-3.96 - 0.525i)T \)
3 \( 1 \)
good5 \( 1 + (11.0 - 19.1i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (82.7 - 47.7i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (18.9 - 10.9i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (63.1 - 109. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 283.T + 8.35e4T^{2} \)
19 \( 1 + 323. iT - 1.30e5T^{2} \)
23 \( 1 + (-198. - 114. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-604. - 1.04e3i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (718. + 414. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 318.T + 1.87e6T^{2} \)
41 \( 1 + (-164. + 284. i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-179. + 103. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.06e3 - 613. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 2.83e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.27e3 - 737. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-936. - 1.62e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (214. + 123. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 4.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.01e3T + 2.83e7T^{2} \)
79 \( 1 + (6.22e3 - 3.59e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-2.87e3 + 1.66e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 1.54e3T + 6.27e7T^{2} \)
97 \( 1 + (2.91e3 + 5.05e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−13.19585997137027279236866837352, −12.41619981090586715407791919589, −11.55758099853809519808067745771, −10.32507975155723527654071573821, −9.079856607755774862273540596159, −7.27841335842118646920997063772, −6.55027704681766202125377076102, −5.32932980754311142041115588838, −3.52647265373802240584645132988, −2.66953790587458197967855659300, 0.69086285654099953892916229887, 3.11671038905536118385210770210, 4.13794017261202606606026002927, 5.60186045230961064832466302572, 6.82095927964212186797563757537, 7.959992529939993092616831042056, 9.802117750262330009741272803467, 10.49635147559482240351848668413, 12.07755658940316765406054916342, 12.70448378311707817495479899689

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