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Properties

Label 2-1560-13.10-c1-0-24
Degree $2$
Conductor $1560$
Sign $-0.0276 + 0.999i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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  + (0.5 + 0.866i)3-s − i·5-s + (−2.23 − 1.29i)7-s + (−0.499 + 0.866i)9-s + (2.68 − 1.54i)11-s + (−1.84 + 3.09i)13-s + (0.866 − 0.5i)15-s + (2.25 − 3.90i)17-s + (−5.29 − 3.05i)19-s − 2.58i·21-s + (2.09 + 3.62i)23-s − 25-s − 0.999·27-s + (−1.72 − 2.98i)29-s − 6.85i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.447i·5-s + (−0.846 − 0.488i)7-s + (−0.166 + 0.288i)9-s + (0.808 − 0.466i)11-s + (−0.512 + 0.858i)13-s + (0.223 − 0.129i)15-s + (0.547 − 0.947i)17-s + (−1.21 − 0.701i)19-s − 0.564i·21-s + (0.436 + 0.756i)23-s − 0.200·25-s − 0.192·27-s + (−0.320 − 0.554i)29-s − 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0276 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0276 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.0276 + 0.999i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ -0.0276 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.164823322\)
\(L(\frac12)\) \(\approx\) \(1.164823322\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 + (1.84 - 3.09i)T \)
good7 \( 1 + (2.23 + 1.29i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 1.54i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.25 + 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.29 + 3.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.09 - 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.72 + 2.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.85iT - 31T^{2} \)
37 \( 1 + (-2.14 + 1.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.39 - 1.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.14 + 8.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 + 6.19T + 53T^{2} \)
59 \( 1 + (8.87 + 5.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.754 - 1.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.37 + 5.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.91 + 5.72i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.59iT - 73T^{2} \)
79 \( 1 - 3.53T + 79T^{2} \)
83 \( 1 + 7.11iT - 83T^{2} \)
89 \( 1 + (4.71 - 2.72i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.9 - 7.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.427524315338310343399821065211, −8.677403999717460589531818720381, −7.60752283045469674680583123827, −6.83103594505931041185980291292, −6.01390076698364767687514173301, −4.91051956104941728016463130876, −4.10556993514922422854359919818, −3.34138829499358880192986744318, −2.11125754001789375050789469158, −0.43326547413524234996568310893, 1.48730154090780867973546142734, 2.70122422119727997763589599656, 3.43648743634715093280408785239, 4.55172725245659932822268561443, 5.89677814453731238868160032321, 6.37301898536675592186629809982, 7.17789900432772121396247601889, 8.063344929747911267111810169871, 8.794563270987508303526658715985, 9.630570921457250444144589297036

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