Properties

Label 2-2520-2520.349-c0-0-5
Degree $2$
Conductor $2520$
Sign $0.819 - 0.573i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
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  + (0.866 + 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 + 0.866i)4-s + (0.258 + 0.965i)5-s + (0.965 − 0.258i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s − 1.00i·9-s + (−0.258 + 0.965i)10-s + (0.965 + 0.258i)12-s + (1.22 − 0.707i)13-s + (−0.499 − 0.866i)14-s + (0.866 + 0.500i)15-s + (−0.5 + 0.866i)16-s + (0.500 − 0.866i)18-s + 1.93·19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.707 − 0.707i)3-s + (0.499 + 0.866i)4-s + (0.258 + 0.965i)5-s + (0.965 − 0.258i)6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s − 1.00i·9-s + (−0.258 + 0.965i)10-s + (0.965 + 0.258i)12-s + (1.22 − 0.707i)13-s + (−0.499 − 0.866i)14-s + (0.866 + 0.500i)15-s + (−0.5 + 0.866i)16-s + (0.500 − 0.866i)18-s + 1.93·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.819 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.523906948\)
\(L(\frac12)\) \(\approx\) \(2.523906948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.93T + T^{2} \)
23 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.065383153417079372109922135540, −7.963744412762845729888346002972, −7.52632917292044583383268679897, −6.90061709815980933083759441381, −6.00779338616854954332594074719, −5.75945234181353388199129728830, −4.07901103066287607152791798540, −3.24864745378742415622737644037, −3.05058794389736640794544758194, −1.63259442846936896478816603866, 1.45088028572874639964028625583, 2.50820806093645604114912952291, 3.46095798864133262589731421900, 4.08711375186388231047851359786, 4.94048191738381606017622126111, 5.74880623684087789000133425742, 6.32638470841691147050429811942, 7.55629787522670288762448998219, 8.557927068896963839810548455493, 9.200973878681045513473804342199

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