Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.985 + 0.172i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.63 − 4.33i)2-s + (−5.51 − 31.5i)4-s + 80.5i·5-s − 216. i·7-s + (−156. − 90.8i)8-s + (348. + 293. i)10-s + 153.·11-s − 945.·13-s + (−938. − 789. i)14-s + (−963. + 347. i)16-s − 2.18e3i·17-s − 719. i·19-s + (2.53e3 − 443. i)20-s + (558. − 664. i)22-s − 2.62e3·23-s + ⋯
L(s)  = 1  + (0.643 − 0.765i)2-s + (−0.172 − 0.985i)4-s + 1.44i·5-s − 1.67i·7-s + (−0.864 − 0.501i)8-s + (1.10 + 0.926i)10-s + 0.382·11-s − 1.55·13-s + (−1.28 − 1.07i)14-s + (−0.940 + 0.339i)16-s − 1.83i·17-s − 0.457i·19-s + (1.41 − 0.248i)20-s + (0.245 − 0.292i)22-s − 1.03·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.985 + 0.172i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.985 + 0.172i)\)
\(L(3)\)  \(\approx\)  \(0.126991 - 1.46392i\)
\(L(\frac12)\)  \(\approx\)  \(0.126991 - 1.46392i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-3.63 + 4.33i)T \)
3 \( 1 \)
good5 \( 1 - 80.5iT - 3.12e3T^{2} \)
7 \( 1 + 216. iT - 1.68e4T^{2} \)
11 \( 1 - 153.T + 1.61e5T^{2} \)
13 \( 1 + 945.T + 3.71e5T^{2} \)
17 \( 1 + 2.18e3iT - 1.41e6T^{2} \)
19 \( 1 + 719. iT - 2.47e6T^{2} \)
23 \( 1 + 2.62e3T + 6.43e6T^{2} \)
29 \( 1 + 155. iT - 2.05e7T^{2} \)
31 \( 1 - 4.90e3iT - 2.86e7T^{2} \)
37 \( 1 - 114.T + 6.93e7T^{2} \)
41 \( 1 - 3.35e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.31e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.40e3T + 2.29e8T^{2} \)
53 \( 1 + 1.96e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.15e4T + 7.14e8T^{2} \)
61 \( 1 - 3.32e4T + 8.44e8T^{2} \)
67 \( 1 + 2.34e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.40e3T + 1.80e9T^{2} \)
73 \( 1 - 4.86e3T + 2.07e9T^{2} \)
79 \( 1 + 1.19e4iT - 3.07e9T^{2} \)
83 \( 1 - 8.39e4T + 3.93e9T^{2} \)
89 \( 1 - 3.68e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.82e4T + 8.58e9T^{2} \)
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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

−12.08535765860415894669924934985, −11.19859907921101692561272300261, −10.28856173998989454393832845522, −9.677824031738943521363046365758, −7.29247714567238249290941596320, −6.78280912202941483216670476093, −4.90637573633911990344134949461, −3.64125628803860245609581050567, −2.46080167187826674060286638360, −0.42484036942573626694974850338, 2.18050458601373869859652385225, 4.17944218368342033859079678271, 5.33344489197337022823246268718, 6.09505476573999811146824969112, 7.938725165237481838331660807467, 8.678185675161964422110066854132, 9.613639581671664193949323543989, 11.88688241361137568782850932667, 12.36808859686092908015714151508, 12.99349368023664272090392735943

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