Properties

Label 2-1560-1.1-c1-0-12
Degree $2$
Conductor $1560$
Sign $1$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 3.43·7-s + 9-s − 2.63·11-s − 13-s + 15-s + 7.84·17-s + 0.794·19-s + 3.43·21-s − 3.43·23-s + 25-s + 27-s − 4.06·29-s + 9.27·31-s − 2.63·33-s + 3.43·35-s − 0.636·37-s − 39-s − 4.63·41-s + 2.41·43-s + 45-s + 5.27·47-s + 4.77·49-s + 7.84·51-s + 11.4·53-s − 2.63·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.29·7-s + 0.333·9-s − 0.795·11-s − 0.277·13-s + 0.258·15-s + 1.90·17-s + 0.182·19-s + 0.748·21-s − 0.715·23-s + 0.200·25-s + 0.192·27-s − 0.755·29-s + 1.66·31-s − 0.458·33-s + 0.580·35-s − 0.104·37-s − 0.160·39-s − 0.724·41-s + 0.367·43-s + 0.149·45-s + 0.769·47-s + 0.682·49-s + 1.09·51-s + 1.57·53-s − 0.355·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.658191725\)
\(L(\frac12)\) \(\approx\) \(2.658191725\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 - 3.43T + 7T^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
17 \( 1 - 7.84T + 17T^{2} \)
19 \( 1 - 0.794T + 19T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
29 \( 1 + 4.06T + 29T^{2} \)
31 \( 1 - 9.27T + 31T^{2} \)
37 \( 1 + 0.636T + 37T^{2} \)
41 \( 1 + 4.63T + 41T^{2} \)
43 \( 1 - 2.41T + 43T^{2} \)
47 \( 1 - 5.27T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 6.41T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 - 8.63T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.472317953188781602547058671510, −8.488176953244069843852140818610, −7.83284790028976280581249569926, −7.41644598289367513556424409482, −6.02078879345166859751202156638, −5.27280077978339403064666770068, −4.50018410272704688600700437853, −3.29129373650755031170999303490, −2.29421508134718927524124795597, −1.26170598600573382275781644085, 1.26170598600573382275781644085, 2.29421508134718927524124795597, 3.29129373650755031170999303490, 4.50018410272704688600700437853, 5.27280077978339403064666770068, 6.02078879345166859751202156638, 7.41644598289367513556424409482, 7.83284790028976280581249569926, 8.488176953244069843852140818610, 9.472317953188781602547058671510

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