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Properties

Label 2-560-140.19-c1-0-14
Degree $2$
Conductor $560$
Sign $0.931 + 0.363i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (0.308 − 0.178i)3-s + (2.20 − 0.358i)5-s + (−2.42 − 1.04i)7-s + (−1.43 + 2.48i)9-s + (3.87 − 2.23i)11-s + 5.47·13-s + (0.617 − 0.504i)15-s + (−0.707 − 1.22i)17-s + (−0.436 + 0.756i)19-s + (−0.936 + 0.109i)21-s + (2.42 − 4.20i)23-s + (4.74 − 1.58i)25-s + 2.09i·27-s + 3.87·29-s + (0.436 + 0.756i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.102i)3-s + (0.987 − 0.160i)5-s + (−0.918 − 0.395i)7-s + (−0.478 + 0.829i)9-s + (1.16 − 0.674i)11-s + 1.51·13-s + (0.159 − 0.130i)15-s + (−0.171 − 0.297i)17-s + (−0.100 + 0.173i)19-s + (−0.204 + 0.0240i)21-s + (0.506 − 0.877i)23-s + (0.948 − 0.316i)25-s + 0.402i·27-s + 0.719·29-s + (0.0783 + 0.135i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.931 + 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74788 - 0.329184i\)
\(L(\frac12)\) \(\approx\) \(1.74788 - 0.329184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.20 + 0.358i)T \)
7 \( 1 + (2.42 + 1.04i)T \)
good3 \( 1 + (-0.308 + 0.178i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3.87 + 2.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.436 - 0.756i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.42 + 4.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 + (-0.436 - 0.756i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.24 + 2.44i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.24iT - 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + (-2.12 - 1.22i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 1.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.43 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.67 - 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.43iT - 71T^{2} \)
73 \( 1 + (2.12 + 3.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.30 - 4.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.41iT - 83T^{2} \)
89 \( 1 + (13.1 + 7.57i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
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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

−10.68750343719259298957561991256, −9.827297199078349572676526922159, −8.837482502420013970718790410050, −8.400795507473842886868745682415, −6.75340256403522884105522117476, −6.31444475142759933878488993449, −5.27857098372760524023419688648, −3.86758478644049725630098141715, −2.78339595743774670812825476485, −1.25081004390823650551627986673, 1.49291509214344025419697770647, 3.02832195546412828516962241982, 3.89907903691284806897764195163, 5.49489480923190633545718814418, 6.42407757734223201292443419422, 6.74568462886707315375260255122, 8.552644336270666256282699563637, 9.149071480249648106081410197932, 9.726002787650305164637111836575, 10.69761025588715522502451796584

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