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Properties

Label 2-3120-3120.1403-c0-0-2
Degree $2$
Conductor $3120$
Sign $-0.584 - 0.811i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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  + 2-s + i·3-s + 4-s + i·5-s + i·6-s + 8-s − 9-s + i·10-s + (−1 + i)11-s + i·12-s − 13-s − 15-s + 16-s − 18-s + i·20-s + ⋯
L(s)  = 1  + 2-s + i·3-s + 4-s + i·5-s + i·6-s + 8-s − 9-s + i·10-s + (−1 + i)11-s + i·12-s − 13-s − 15-s + 16-s − 18-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.035174883\)
\(L(\frac12)\) \(\approx\) \(2.035174883\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.496238997599421458798470464748, −8.157128188556396190875771063384, −7.43510796139350596589494563125, −6.86952089080086558783993746980, −5.82555661494865851012457915834, −5.24602744550080692119175667667, −4.44247916904605191733255600988, −3.76667019542062387733665902799, −2.65158049332213273743033510181, −2.37127365801496944899371412359, 0.843638944677589109054227339574, 2.15435167803244157224935064320, 2.81466593245109444833348032937, 3.93795290870063400545831205417, 4.97163892519985443812836991496, 5.54786378194440171181222285224, 6.09342508400412306894340505145, 7.16841779185737730785728503484, 7.70426195352367840806056064916, 8.356767048893687183405065412887

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