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Properties

Label 2-570-95.8-c1-0-12
Degree $2$
Conductor $570$
Sign $0.699 + 0.714i$
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  + (0.965 − 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (−0.743 + 2.10i)5-s + (−0.499 − 0.866i)6-s + (2.46 − 2.46i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.172 + 2.22i)10-s + 3.74·11-s + (−0.707 − 0.707i)12-s + (−1.91 − 0.512i)13-s + (1.74 − 3.01i)14-s + (2.22 + 0.172i)15-s + (0.500 − 0.866i)16-s + (1.16 + 4.33i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.332 + 0.943i)5-s + (−0.204 − 0.353i)6-s + (0.930 − 0.930i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.0544 + 0.705i)10-s + 1.12·11-s + (−0.204 − 0.204i)12-s + (−0.530 − 0.142i)13-s + (0.465 − 0.806i)14-s + (0.575 + 0.0444i)15-s + (0.125 − 0.216i)16-s + (0.281 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05375 - 0.863103i\)
\(L(\frac12)\) \(\approx\) \(2.05375 - 0.863103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.743 - 2.10i)T \)
19 \( 1 + (-2.18 + 3.77i)T \)
good7 \( 1 + (-2.46 + 2.46i)T - 7iT^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 + (1.91 + 0.512i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.16 - 4.33i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-2.08 + 7.79i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.22 - 3.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.07iT - 31T^{2} \)
37 \( 1 + (1.80 + 1.80i)T + 37iT^{2} \)
41 \( 1 + (-4.07 - 2.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.11 - 1.63i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.981 - 0.263i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.75 + 1.81i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.04 - 5.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.36 - 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.342 + 1.27i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (12.6 + 7.27i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.73 - 2.60i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.64 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.69 - 8.69i)T + 83iT^{2} \)
89 \( 1 + (3.14 + 5.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.07 + 0.824i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.81981317386868416978536313383, −10.20566479859809609324504580885, −8.697790069645728073099319473496, −7.62503425916926359057034964791, −6.94810381517334864728707604189, −6.25316253483406386056898074749, −4.84242301943407237624862061267, −3.96900418054334698306953607954, −2.75943390477507387061934381048, −1.29615352547006729994206026151, 1.64112262407425346846606939828, 3.33990369452630305977312996081, 4.46511002877726651173739068306, 5.17182580885163675320309213663, 5.87108545576797528685382453133, 7.31482037226129323993431877541, 8.197614760336012624810349596397, 9.144371483897883814348948056272, 9.756773270020914935219331095608, 11.33537629474219734139219519766

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