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Properties

Label 2-560-7.2-c1-0-0
Degree $2$
Conductor $560$
Sign $0.845 - 0.533i$
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  + (−1.64 − 2.84i)3-s + (−0.5 + 0.866i)5-s + (−2.64 − 0.0641i)7-s + (−3.91 + 6.77i)9-s + (2.91 + 5.04i)11-s + 2.75·13-s + 3.28·15-s + (−1 − 1.73i)17-s + (−0.378 + 0.654i)19-s + (4.16 + 7.64i)21-s + (−0.266 + 0.462i)23-s + (−0.499 − 0.866i)25-s + 15.8·27-s − 0.823·29-s + (−1.28 − 2.23i)31-s + ⋯
L(s)  = 1  + (−0.949 − 1.64i)3-s + (−0.223 + 0.387i)5-s + (−0.999 − 0.0242i)7-s + (−1.30 + 2.25i)9-s + (0.877 + 1.52i)11-s + 0.764·13-s + 0.849·15-s + (−0.242 − 0.420i)17-s + (−0.0867 + 0.150i)19-s + (0.909 + 1.66i)21-s + (−0.0556 + 0.0963i)23-s + (−0.0999 − 0.173i)25-s + 3.05·27-s − 0.152·29-s + (−0.231 − 0.401i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.845 - 0.533i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620353 + 0.179295i\)
\(L(\frac12)\) \(\approx\) \(0.620353 + 0.179295i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.64 + 0.0641i)T \)
good3 \( 1 + (1.64 + 2.84i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.91 - 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.378 - 0.654i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.266 - 0.462i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.823T + 29T^{2} \)
31 \( 1 + (1.28 + 2.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.37 - 4.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 + 0.710T + 43T^{2} \)
47 \( 1 + (6.44 - 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.20 - 7.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.70 - 8.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.93 - 10.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1.75 + 3.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.75 - 8.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.71T + 83T^{2} \)
89 \( 1 + (0.878 - 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.12049823159638223466350324019, −10.11684565693924715964508356426, −9.046048908668423960283222561134, −7.76065599539411876058034291894, −7.02953871089351993909070759967, −6.51437605400881093207340172220, −5.71121927928558312000667255726, −4.25763233255412065159024056702, −2.62426234572303243918759667088, −1.32140756762208420421863705226, 0.45938387455822089526707781624, 3.52210124424857656561903401631, 3.77385364016134474081111336543, 5.10759435377205219947727066844, 6.01451699625565853755512659875, 6.55411631493811211580056382443, 8.544697824683611794765917453840, 9.039096125036247933235447872891, 9.856971911995505327560631448322, 10.78536299113617858014902414452

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