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Properties

Label 2-560-140.27-c2-0-21
Degree $2$
Conductor $560$
Sign $0.298 - 0.954i$
Analytic cond. $15.2588$
Root an. cond. $3.90626$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.74 + 3.74i)3-s + (3.76 − 3.29i)5-s + (−0.0261 + 6.99i)7-s − 19.0i·9-s − 5.33i·11-s + (13.8 + 13.8i)13-s + (−1.77 + 26.4i)15-s + (12.8 − 12.8i)17-s − 25.2i·19-s + (−26.1 − 26.3i)21-s + (12.2 + 12.2i)23-s + (3.34 − 24.7i)25-s + (37.6 + 37.6i)27-s + 13.3i·29-s + 46.4·31-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)3-s + (0.752 − 0.658i)5-s + (−0.00372 + 0.999i)7-s − 2.11i·9-s − 0.485i·11-s + (1.06 + 1.06i)13-s + (−0.118 + 1.76i)15-s + (0.758 − 0.758i)17-s − 1.33i·19-s + (−1.24 − 1.25i)21-s + (0.534 + 0.534i)23-s + (0.133 − 0.990i)25-s + (1.39 + 1.39i)27-s + 0.460i·29-s + 1.49·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.298 - 0.954i$
Analytic conductor: \(15.2588\)
Root analytic conductor: \(3.90626\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1),\ 0.298 - 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.381926395\)
\(L(\frac12)\) \(\approx\) \(1.381926395\)
\(L(2)\) 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 \( 1 + (-3.76 + 3.29i)T \)
7 \( 1 + (0.0261 - 6.99i)T \)
good3 \( 1 + (3.74 - 3.74i)T - 9iT^{2} \)
11 \( 1 + 5.33iT - 121T^{2} \)
13 \( 1 + (-13.8 - 13.8i)T + 169iT^{2} \)
17 \( 1 + (-12.8 + 12.8i)T - 289iT^{2} \)
19 \( 1 + 25.2iT - 361T^{2} \)
23 \( 1 + (-12.2 - 12.2i)T + 529iT^{2} \)
29 \( 1 - 13.3iT - 841T^{2} \)
31 \( 1 - 46.4T + 961T^{2} \)
37 \( 1 + (-6.06 - 6.06i)T + 1.36e3iT^{2} \)
41 \( 1 - 80.3iT - 1.68e3T^{2} \)
43 \( 1 + (40.6 + 40.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-19.3 - 19.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (34.8 - 34.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 58.6iT - 3.48e3T^{2} \)
61 \( 1 - 14.4iT - 3.72e3T^{2} \)
67 \( 1 + (36.9 - 36.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 24.3iT - 5.04e3T^{2} \)
73 \( 1 + (-30.9 - 30.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 135.T + 6.24e3T^{2} \)
83 \( 1 + (-22.0 + 22.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 12.2T + 7.92e3T^{2} \)
97 \( 1 + (-46.1 + 46.1i)T - 9.40e3iT^{2} \)
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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

−10.81262095777366925432529818194, −9.733347137191006660508637958201, −9.240475712116133501258343619595, −8.543089522996010720478456018063, −6.60695140282805717065059180873, −5.94339569444932861497921173143, −5.13596057822287168281251910643, −4.50412546638635204975368833628, −3.02003146071992651315706306564, −1.06056764220419738306038968513, 0.832626867868080095505607421474, 1.83466532893422729369447905456, 3.51490740789041723930063090696, 5.12709434193877815138755466468, 6.11095801934526627090019853160, 6.49378353770497181755732464564, 7.55255923018238465381715422556, 8.165021184564233133350946984145, 10.02285175409569267010704820072, 10.49993155078154784624667586816

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