Properties

Label 2-3120-5.4-c1-0-51
Degree $2$
Conductor $3120$
Sign $0.894 - 0.447i$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (2 − i)5-s + 4i·7-s − 9-s + 6·11-s i·13-s + (1 + 2i)15-s − 4i·17-s + 2·19-s − 4·21-s − 6i·23-s + (3 − 4i)25-s i·27-s + 10·29-s − 4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.894 − 0.447i)5-s + 1.51i·7-s − 0.333·9-s + 1.80·11-s − 0.277i·13-s + (0.258 + 0.516i)15-s − 0.970i·17-s + 0.458·19-s − 0.872·21-s − 1.25i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s + 1.85·29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.618500213\)
\(L(\frac12)\) \(\approx\) \(2.618500213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (-2 + i)T \)
13 \( 1 + iT \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.958040616746122333111201474991, −8.424829985350713828616220005511, −7.09351890909329506586324070245, −6.21263921756240562714508946679, −5.76832631751107995023179159508, −4.92785139816646120842257796339, −4.22477599879419634678394927328, −2.96158915716007055894373938844, −2.27845855051693657560688095724, −1.01865613135659887529208141765, 1.18695580078384063840761271952, 1.57864541294792856600228268624, 3.05744024629474237944457919649, 3.85582733959903566998354501762, 4.65945338962121549324356948746, 5.96078293697032060894740412650, 6.42666367018735506774205082162, 7.05381595381731855820387239897, 7.66689760879511475984574926285, 8.663198519662771624716587161983

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