Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.639 - 0.768i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 − 3.76i)2-s + (−12.3 + 10.2i)4-s + 2.66·5-s − 88.8i·7-s + (55.2 + 32.3i)8-s + (−3.61 − 10.0i)10-s + 72.9i·11-s − 173.·13-s + (−334. + 120. i)14-s + (46.7 − 251. i)16-s − 423.·17-s + 153. i·19-s + (−32.7 + 27.2i)20-s + (274. − 99.1i)22-s + 723. i·23-s + ⋯
L(s)  = 1  + (−0.339 − 0.940i)2-s + (−0.768 + 0.639i)4-s + 0.106·5-s − 1.81i·7-s + (0.862 + 0.505i)8-s + (−0.0361 − 0.100i)10-s + 0.602i·11-s − 1.02·13-s + (−1.70 + 0.616i)14-s + (0.182 − 0.983i)16-s − 1.46·17-s + 0.424i·19-s + (−0.0818 + 0.0680i)20-s + (0.567 − 0.204i)22-s + 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.639 - 0.768i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.639 - 0.768i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.127482 + 0.271787i\)
\(L(\frac12)\)  \(\approx\)  \(0.127482 + 0.271787i\)
\(L(3)\)   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 \( 1 + (1.35 + 3.76i)T \)
3 \( 1 \)
good5 \( 1 - 2.66T + 625T^{2} \)
7 \( 1 + 88.8iT - 2.40e3T^{2} \)
11 \( 1 - 72.9iT - 1.46e4T^{2} \)
13 \( 1 + 173.T + 2.85e4T^{2} \)
17 \( 1 + 423.T + 8.35e4T^{2} \)
19 \( 1 - 153. iT - 1.30e5T^{2} \)
23 \( 1 - 723. iT - 2.79e5T^{2} \)
29 \( 1 - 366.T + 7.07e5T^{2} \)
31 \( 1 + 642. iT - 9.23e5T^{2} \)
37 \( 1 + 535.T + 1.87e6T^{2} \)
41 \( 1 + 2.14e3T + 2.82e6T^{2} \)
43 \( 1 + 615. iT - 3.41e6T^{2} \)
47 \( 1 - 330. iT - 4.87e6T^{2} \)
53 \( 1 - 4.62e3T + 7.89e6T^{2} \)
59 \( 1 + 3.46e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.64e3T + 1.38e7T^{2} \)
67 \( 1 + 6.67e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.44e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.73e3T + 2.83e7T^{2} \)
79 \( 1 + 2.08e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.29e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.70e3T + 6.27e7T^{2} \)
97 \( 1 - 6.97e3T + 8.85e7T^{2} \)
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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

−12.20099964591107890578050390895, −11.12250149567781120261284885611, −10.19407298357460758407028257078, −9.483887588484167655949820820104, −7.88206519632996341921552737479, −7.02463619371487229361843168616, −4.74977422036161001394435339382, −3.73691590202222030758066679580, −1.86256621160982315710344715938, −0.14354681859381375727616163243, 2.38321031305154646931017364456, 4.75580261610172361196528128125, 5.82381805014601863785296526215, 6.86346671767599495128439089294, 8.485311092612526808056998213742, 8.943211566468646649600575816031, 10.17856392219750353491850988230, 11.62730189971921769968517308274, 12.71779377391355892325717371068, 13.86066177551823994132962936765

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