Properties

Label 2-3120-5.4-c1-0-12
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 − 2i·7-s − 9-s − 2·11-s + i·13-s + (−1 + 2i)15-s + 2i·17-s − 4·19-s + 2·21-s + (3 + 4i)25-s i·27-s − 4·29-s − 8·31-s − 2i·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.894 + 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s + 0.277i·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s − 0.917·19-s + 0.436·21-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 0.742·29-s − 1.43·31-s − 0.348i·33-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

\(L(1)\) \(\approx\) \(1.020274996\)
\(L(\frac12)\) \(\approx\) \(1.020274996\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.133268811439705425744653861659, −8.344736712617183097635433907133, −7.45040171591637541988342342090, −6.71509816029360762341239124059, −5.94947025571532300547259944484, −5.21686885616493223095949956927, −4.31890707998252421772462088689, −3.51500931558475101007930422432, −2.53329523386891776614274580468, −1.52987831900190524830934529347, 0.28491494573742795891265509846, 1.83546881909555737139101018036, 2.32225797222956469181091582995, 3.43088109584937124432054037674, 4.72234377214669647550459857644, 5.55896621300738060861822604438, 5.86624795008074368619694933757, 6.89687004795153531334743827776, 7.57750170309230984537305192346, 8.603798841262593642195515003299

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