Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s i·5-s − 6-s i·8-s − 9-s + 10-s i·12-s + 15-s + 16-s i·18-s + i·20-s + 24-s − 25-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s i·5-s − 6-s i·8-s − 9-s + 10-s i·12-s + 15-s + 16-s i·18-s + i·20-s + 24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(0\)
character  :  $\chi_{120} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 120,\ (\ :0),\ -i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5437162785\)
\(L(\frac12)\)  \(\approx\)  \(0.5437162785\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + iT \)
good7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−14.21271855810921919646769460812, −13.19667196832582720061135768379, −12.09747784861115319834900421262, −10.61437846076570269693066417743, −9.388643639095434605845356081616, −8.817852444608721014831468079651, −7.63338202461196921745610868863, −5.91048570812207665106258742322, −4.97657542880787386510676331574, −3.86980282416878206705418894327, 2.15313362465143136876203290842, 3.52472146273302481464633973046, 5.56402930318338092618140678491, 6.96322967351268565631502757622, 8.108885684803283632899512093538, 9.386450812199118293810282841415, 10.68268464629724600508738886889, 11.41393770282462266589397115446, 12.39747538269979838031491661615, 13.33009494655484601976149952531

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