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Properties

Label 2-560-35.33-c1-0-2
Degree $2$
Conductor $560$
Sign $-0.699 - 0.714i$
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  + (−2.20 − 0.590i)3-s + (−0.352 + 2.20i)5-s + (1.22 − 2.34i)7-s + (1.90 + 1.10i)9-s + (0.0644 + 0.111i)11-s + (−0.748 + 0.748i)13-s + (2.07 − 4.65i)15-s + (0.613 − 2.28i)17-s + (−3.81 + 6.59i)19-s + (−4.08 + 4.44i)21-s + (−4.11 + 1.10i)23-s + (−4.75 − 1.55i)25-s + (1.28 + 1.28i)27-s + 0.163i·29-s + (−8.43 + 4.86i)31-s + ⋯
L(s)  = 1  + (−1.27 − 0.340i)3-s + (−0.157 + 0.987i)5-s + (0.462 − 0.886i)7-s + (0.636 + 0.367i)9-s + (0.0194 + 0.0336i)11-s + (−0.207 + 0.207i)13-s + (0.536 − 1.20i)15-s + (0.148 − 0.555i)17-s + (−0.874 + 1.51i)19-s + (−0.890 + 0.970i)21-s + (−0.857 + 0.229i)23-s + (−0.950 − 0.310i)25-s + (0.246 + 0.246i)27-s + 0.0303i·29-s + (−1.51 + 0.874i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.699 - 0.714i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128622 + 0.305952i\)
\(L(\frac12)\) \(\approx\) \(0.128622 + 0.305952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.352 - 2.20i)T \)
7 \( 1 + (-1.22 + 2.34i)T \)
good3 \( 1 + (2.20 + 0.590i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.0644 - 0.111i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.748 - 0.748i)T - 13iT^{2} \)
17 \( 1 + (-0.613 + 2.28i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.81 - 6.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.11 - 1.10i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.163iT - 29T^{2} \)
31 \( 1 + (8.43 - 4.86i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0241 - 0.0900i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.9iT - 41T^{2} \)
43 \( 1 + (2.76 + 2.76i)T + 43iT^{2} \)
47 \( 1 + (2.23 - 0.597i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.96 - 7.32i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.13 - 7.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.5 + 6.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.84 + 1.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.50T + 71T^{2} \)
73 \( 1 + (-1.80 - 0.483i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.48 - 4.89i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.43 + 8.43i)T - 83iT^{2} \)
89 \( 1 + (-3.37 + 5.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.61 + 9.61i)T + 97iT^{2} \)
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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

−10.96725432729010853298728448008, −10.58844960639827390510996597799, −9.685449423901640823284696910670, −8.100117034283209093651637856498, −7.33027958659002496232027569724, −6.54102435191444094734352916123, −5.77382535938195291578906297115, −4.59592078142989096975316155611, −3.47385452934216855691707037489, −1.66452300834571438441033624923, 0.21863691256913937100954524482, 2.08314950842103914521051106440, 4.07166196820101741434142344903, 5.02838992875872683735500214470, 5.56775230951811491145887297746, 6.50059973499625474586699008279, 7.892495879199453453075886190158, 8.749388738417308803938784997388, 9.529255448018654411515743550647, 10.71320659747628726884865671566

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