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Properties

Label 2-560-112.109-c1-0-41
Degree $2$
Conductor $560$
Sign $-0.606 + 0.795i$
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  + (−1.13 + 0.838i)2-s + (−2.01 + 0.539i)3-s + (0.592 − 1.91i)4-s + (0.965 + 0.258i)5-s + (1.83 − 2.30i)6-s + (0.308 − 2.62i)7-s + (0.926 + 2.67i)8-s + (1.16 − 0.670i)9-s + (−1.31 + 0.515i)10-s + (−0.473 − 1.76i)11-s + (−0.163 + 4.16i)12-s + (0.888 + 0.888i)13-s + (1.85 + 3.25i)14-s − 2.08·15-s + (−3.29 − 2.26i)16-s + (−3.54 + 6.14i)17-s + ⋯
L(s)  = 1  + (−0.805 + 0.593i)2-s + (−1.16 + 0.311i)3-s + (0.296 − 0.955i)4-s + (0.431 + 0.115i)5-s + (0.750 − 0.939i)6-s + (0.116 − 0.993i)7-s + (0.327 + 0.944i)8-s + (0.386 − 0.223i)9-s + (−0.416 + 0.163i)10-s + (−0.142 − 0.532i)11-s + (−0.0471 + 1.20i)12-s + (0.246 + 0.246i)13-s + (0.495 + 0.868i)14-s − 0.537·15-s + (−0.824 − 0.566i)16-s + (−0.860 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.606 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0492461 - 0.0994838i\)
\(L(\frac12)\) \(\approx\) \(0.0492461 - 0.0994838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.838i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-0.308 + 2.62i)T \)
good3 \( 1 + (2.01 - 0.539i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.473 + 1.76i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.888 - 0.888i)T + 13iT^{2} \)
17 \( 1 + (3.54 - 6.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.611 + 2.28i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (6.40 - 3.69i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.60 + 4.60i)T + 29iT^{2} \)
31 \( 1 + (-0.980 + 1.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.76 + 1.54i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.87iT - 41T^{2} \)
43 \( 1 + (0.178 - 0.178i)T - 43iT^{2} \)
47 \( 1 + (4.96 + 8.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.30 + 4.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.77 + 10.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.65 + 6.18i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (13.1 - 3.51i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 + (-0.795 - 0.459i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.64 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.11 - 1.11i)T + 83iT^{2} \)
89 \( 1 + (0.683 - 0.394i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.24T + 97T^{2} \)
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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

−10.37032830509051689237860950761, −9.846207435743744530669816814291, −8.627989444044253072263268077299, −7.77366110469340224991409298135, −6.59547769712400439184603179459, −6.08490846739710404657041562360, −5.16518842925963890030449227396, −3.99836922847954798559270684002, −1.77113443861332710776896815864, −0.093215201820837211372330465479, 1.67479248138535250787258727535, 2.86301788887506448352850098965, 4.63421098656609956830495311510, 5.67316177118481317502301228927, 6.55815657874407215302789796368, 7.51026714170922266094635793731, 8.693036504577694334944763783782, 9.341671120762339842957268515179, 10.35164888207093057525079848468, 11.05028912677064788211261703286

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