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Properties

Label 2-1560-8.5-c1-0-7
Degree $2$
Conductor $1560$
Sign $-i$
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  − 1.41·2-s − i·3-s + 2.00·4-s − i·5-s + 1.41i·6-s − 3.41·7-s − 2.82·8-s − 9-s + 1.41i·10-s + 0.828i·11-s − 2.00i·12-s + i·13-s + 4.82·14-s − 15-s + 4.00·16-s + 1.41·17-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.577i·3-s + 1.00·4-s − 0.447i·5-s + 0.577i·6-s − 1.29·7-s − 1.00·8-s − 0.333·9-s + 0.447i·10-s + 0.249i·11-s − 0.577i·12-s + 0.277i·13-s + 1.29·14-s − 0.258·15-s + 1.00·16-s + 0.342·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2447297264\)
\(L(\frac12)\) \(\approx\) \(0.2447297264\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
13 \( 1 - iT \)
good7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 8.24iT - 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 + 1.75iT - 29T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 - 10.4iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 0.828iT - 43T^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 - 8.82iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 6.82iT - 67T^{2} \)
71 \( 1 + 0.828T + 71T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 - 9.65T + 79T^{2} \)
83 \( 1 + 3.65iT - 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−9.719986221315573872101529146037, −8.780337115290860035191178650903, −8.215956400480549538296526482523, −7.16743392459159329473741598557, −6.63936620394605906386408731069, −5.96631532074561043772090171438, −4.71243635995629284807116286703, −3.25293868261969331399702894511, −2.45252530245709299287790354234, −1.10285649663368728261398752260, 0.14547185425451951508320903358, 1.96400827553388322132464803786, 3.28152628471359816703317899874, 3.63788087916255792478830394200, 5.39138410246780515870443764842, 6.23359293566417105726757317582, 6.73174695235269875842294076626, 7.951727154802383620353614167208, 8.358084827851294555459682833634, 9.699088590314198988156950072227

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