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Properties

Label 2-120-120.29-c0-0-1
Degree $2$
Conductor $120$
Sign $i$
Analytic cond. $0.0598878$
Root an. cond. $0.244719$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $i$
Analytic conductor: \(0.0598878\)
Root analytic conductor: \(0.244719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5437162785\)
\(L(\frac12)\) \(\approx\) \(0.5437162785\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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