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Properties

Label 2-570-57.8-c1-0-17
Degree $2$
Conductor $570$
Sign $-0.517 + 0.855i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (0.5 − 0.866i)2-s + (−0.224 + 1.71i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.37 + 1.05i)6-s − 1.74·7-s − 0.999·8-s + (−2.89 − 0.772i)9-s + (−0.866 + 0.499i)10-s − 4.48i·11-s + (1.59 − 0.663i)12-s + (3.14 − 1.81i)13-s + (−0.871 + 1.51i)14-s + (1.05 − 1.37i)15-s + (−0.5 + 0.866i)16-s + (−5.72 − 3.30i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.129 + 0.991i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.561 + 0.430i)6-s − 0.659·7-s − 0.353·8-s + (−0.966 − 0.257i)9-s + (−0.273 + 0.158i)10-s − 1.35i·11-s + (0.461 − 0.191i)12-s + (0.872 − 0.503i)13-s + (−0.232 + 0.403i)14-s + (0.271 − 0.354i)15-s + (−0.125 + 0.216i)16-s + (−1.38 − 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.517 + 0.855i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.517 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448272 - 0.795207i\)
\(L(\frac12)\) \(\approx\) \(0.448272 - 0.795207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.224 - 1.71i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-3.74 + 2.22i)T \)
good7 \( 1 + 1.74T + 7T^{2} \)
11 \( 1 + 4.48iT - 11T^{2} \)
13 \( 1 + (-3.14 + 1.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.72 + 3.30i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.69 - 0.977i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.271 - 0.469i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 - 0.251iT - 37T^{2} \)
41 \( 1 + (2.28 - 3.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.13 - 3.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.28 - 0.743i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0649 - 0.112i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.22 + 5.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.04 - 1.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.40 - 4.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.78 + 8.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.51 - 6.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.04 + 2.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (-2.10 - 3.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.01 - 5.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.62450520122104583112333052846, −9.592323135205492594489375979197, −8.987944435080957307439586647883, −8.091580513532495284449862165362, −6.45182700041129172039565538986, −5.64847187819208937978649202209, −4.61350735865494528831775011820, −3.59251706540123816611377621440, −2.88826655415992120042533464423, −0.46070637837515691766471314476, 1.87933457747757673227949067939, 3.37517364952518106821149838663, 4.51933160408621925210493697613, 5.80943453027692090662805688470, 6.75468197181785663548254397809, 7.08656449501369545991732862901, 8.227000637424844273596680927750, 8.954396232643499755450538603844, 10.18113989326231701696979538938, 11.24765043097199211341605768140

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