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Properties

Label 2-560-112.109-c1-0-25
Degree $2$
Conductor $560$
Sign $0.605 - 0.795i$
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  + (1.39 + 0.236i)2-s + (−2.75 + 0.737i)3-s + (1.88 + 0.659i)4-s + (0.965 + 0.258i)5-s + (−4.01 + 0.377i)6-s + (0.122 − 2.64i)7-s + (2.47 + 1.36i)8-s + (4.43 − 2.56i)9-s + (1.28 + 0.589i)10-s + (0.994 + 3.71i)11-s + (−5.68 − 0.422i)12-s + (1.34 + 1.34i)13-s + (0.795 − 3.65i)14-s − 2.85·15-s + (3.13 + 2.49i)16-s + (1.37 − 2.37i)17-s + ⋯
L(s)  = 1  + (0.985 + 0.167i)2-s + (−1.58 + 0.425i)3-s + (0.944 + 0.329i)4-s + (0.431 + 0.115i)5-s + (−1.63 + 0.154i)6-s + (0.0461 − 0.998i)7-s + (0.875 + 0.482i)8-s + (1.47 − 0.854i)9-s + (0.406 + 0.186i)10-s + (0.299 + 1.11i)11-s + (−1.64 − 0.122i)12-s + (0.373 + 0.373i)13-s + (0.212 − 0.977i)14-s − 0.736·15-s + (0.782 + 0.622i)16-s + (0.332 − 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.605 - 0.795i$
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: \((2,\ 560,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66619 + 0.825646i\)
\(L(\frac12)\) \(\approx\) \(1.66619 + 0.825646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.236i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-0.122 + 2.64i)T \)
good3 \( 1 + (2.75 - 0.737i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.994 - 3.71i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.34 - 1.34i)T + 13iT^{2} \)
17 \( 1 + (-1.37 + 2.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.643 - 2.40i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.14 + 0.661i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.59 - 3.59i)T + 29iT^{2} \)
31 \( 1 + (3.71 - 6.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.72 - 2.06i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.99iT - 41T^{2} \)
43 \( 1 + (1.43 - 1.43i)T - 43iT^{2} \)
47 \( 1 + (0.326 + 0.564i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.33 + 12.4i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.74 + 13.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.75 + 6.56i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.104 + 0.0279i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 + (6.31 + 3.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.57 - 2.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.732 - 0.732i)T + 83iT^{2} \)
89 \( 1 + (-8.80 + 5.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.96T + 97T^{2} \)
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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

−11.00667478771233378054694917053, −10.35601798306815305567667529351, −9.605576501964060382419594770081, −7.77412580473619139567221218823, −6.68347365410641537784801598021, −6.42630670001720073603814700080, −5.06094391944149720367267641280, −4.67711374165472066765755574603, −3.53078065794654619533327651641, −1.50438918609269937880846851748, 1.11786801419427731590044160173, 2.63178337164583893830198522050, 4.20498806744601191704851307213, 5.44653069147689812953577637917, 5.86523290902135600717397286249, 6.38026706297517642966084774651, 7.61682014024552555624496112197, 8.933568750338640456171215539215, 10.23473881862191228202644794263, 11.04380533513216167673536197672

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