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Properties

Label 2-560-35.17-c1-0-12
Degree $2$
Conductor $560$
Sign $0.361 + 0.932i$
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  + (−1.66 + 0.447i)3-s + (2.21 + 0.298i)5-s + (−0.900 − 2.48i)7-s + (−0.0138 + 0.00800i)9-s + (1.10 − 1.91i)11-s + (−1.91 − 1.91i)13-s + (−3.83 + 0.491i)15-s + (−1.12 − 4.19i)17-s + (1.80 + 3.12i)19-s + (2.61 + 3.74i)21-s + (−0.375 − 0.100i)23-s + (4.82 + 1.32i)25-s + (3.68 − 3.68i)27-s − 6.62i·29-s + (−0.897 − 0.518i)31-s + ⋯
L(s)  = 1  + (−0.963 + 0.258i)3-s + (0.991 + 0.133i)5-s + (−0.340 − 0.940i)7-s + (−0.00462 + 0.00266i)9-s + (0.333 − 0.578i)11-s + (−0.530 − 0.530i)13-s + (−0.989 + 0.127i)15-s + (−0.272 − 1.01i)17-s + (0.413 + 0.716i)19-s + (0.570 + 0.817i)21-s + (−0.0783 − 0.0209i)23-s + (0.964 + 0.264i)25-s + (0.708 − 0.708i)27-s − 1.22i·29-s + (−0.161 − 0.0930i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.361 + 0.932i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818353 - 0.560336i\)
\(L(\frac12)\) \(\approx\) \(0.818353 - 0.560336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.298i)T \)
7 \( 1 + (0.900 + 2.48i)T \)
good3 \( 1 + (1.66 - 0.447i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.10 + 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.91 + 1.91i)T + 13iT^{2} \)
17 \( 1 + (1.12 + 4.19i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.80 - 3.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.375 + 0.100i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.62iT - 29T^{2} \)
31 \( 1 + (0.897 + 0.518i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.84 + 6.88i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.03iT - 41T^{2} \)
43 \( 1 + (-8.37 + 8.37i)T - 43iT^{2} \)
47 \( 1 + (5.52 + 1.48i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.62 - 6.07i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.71 - 6.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.84 + 4.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.0 + 2.70i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.39T + 71T^{2} \)
73 \( 1 + (13.7 - 3.69i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.38 - 3.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.77 + 7.77i)T + 83iT^{2} \)
89 \( 1 + (-5.20 - 9.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.1 - 12.1i)T - 97iT^{2} \)
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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

−10.56069429030274594190762367193, −9.940538477918235276473294287805, −9.123053386079599578129202801038, −7.75306652187342523349252320116, −6.79029931634539565948854673943, −5.86879924957900374891454691855, −5.25428444344998229150147665241, −3.99667430428277879590847815263, −2.57024448921364876521442057245, −0.65624005199564301348850191501, 1.57736274467880126004768393910, 2.86479670199984826592747302484, 4.69592207183623465857886695332, 5.50568508014642337835891479680, 6.35458842435483319093436332861, 6.89459988025208741033842273904, 8.479056506254832766383527341948, 9.316928447169775080621883070697, 9.957378243246674687354568315838, 11.08966763388955225163111714028

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