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Properties

Label 2-1560-1560.1403-c0-0-2
Degree $2$
Conductor $1560$
Sign $-0.229 - 0.973i$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
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  + (−0.382 − 0.923i)2-s + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)5-s + (0.382 − 0.923i)6-s + (0.923 + 0.382i)8-s + 1.00i·9-s + 10-s − 1.84·11-s − 12-s + (−0.707 + 0.707i)13-s + (−0.923 + 0.382i)15-s − i·16-s + (0.923 − 0.382i)18-s + (−0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)5-s + (0.382 − 0.923i)6-s + (0.923 + 0.382i)8-s + 1.00i·9-s + 10-s − 1.84·11-s − 12-s + (−0.707 + 0.707i)13-s + (−0.923 + 0.382i)15-s − i·16-s + (0.923 − 0.382i)18-s + (−0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6414203393\)
\(L(\frac12)\) \(\approx\) \(0.6414203393\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + 1.84T + T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.765iT - T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.84iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
89 \( 1 - 1.84iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.952539221821536974152416446205, −9.303170051417959421105743614806, −8.291131329799431161473280728128, −7.76147596821595895636151140999, −7.06260493031757529587169937126, −5.42265236079829842758044838917, −4.56689322865299826099076952137, −3.69885892486212747686099568515, −2.72010490759924103216621579801, −2.28423962520395933815014765202, 0.50116809066366794890095799437, 2.06247559250676054708604681768, 3.35002569589319896260021749475, 4.71263030396386845163028434845, 5.30774906205377968519676344973, 6.25365602202958854978562653834, 7.45693842169225021191702170417, 7.76489383967042580730359538971, 8.347094738593597890012457271960, 9.114980989339304345571118340504

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