Properties

Label 2-570-57.56-c1-0-16
Degree $2$
Conductor $570$
Sign $-0.437 + 0.899i$
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  − 2-s + (0.209 − 1.71i)3-s + 4-s + i·5-s + (−0.209 + 1.71i)6-s + 0.264·7-s − 8-s + (−2.91 − 0.719i)9-s i·10-s + 2i·11-s + (0.209 − 1.71i)12-s − 5.43i·13-s − 0.264·14-s + (1.71 + 0.209i)15-s + 16-s − 3.28i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.120 − 0.992i)3-s + 0.5·4-s + 0.447i·5-s + (−0.0854 + 0.701i)6-s + 0.0999·7-s − 0.353·8-s + (−0.970 − 0.239i)9-s − 0.316i·10-s + 0.603i·11-s + (0.0603 − 0.496i)12-s − 1.50i·13-s − 0.0707·14-s + (0.443 + 0.0540i)15-s + 0.250·16-s − 0.796i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467224 - 0.746656i\)
\(L(\frac12)\) \(\approx\) \(0.467224 - 0.746656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.209 + 1.71i)T \)
5 \( 1 - iT \)
19 \( 1 + (-1.41 + 4.12i)T \)
good7 \( 1 - 0.264T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 5.43iT - 13T^{2} \)
17 \( 1 + 3.28iT - 17T^{2} \)
23 \( 1 + 3.96iT - 23T^{2} \)
29 \( 1 - 0.418T + 29T^{2} \)
31 \( 1 + 3.56iT - 31T^{2} \)
37 \( 1 + 0.307iT - 37T^{2} \)
41 \( 1 + 2.15T + 41T^{2} \)
43 \( 1 - 1.71T + 43T^{2} \)
47 \( 1 + 5.71iT - 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 9.18T + 61T^{2} \)
67 \( 1 - 0.666iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 8.08iT - 79T^{2} \)
83 \( 1 + 16.7iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 1.47iT - 97T^{2} \)
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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

−10.45675653243975267569808135310, −9.532027114289653926260719150143, −8.545683495572799147731698445947, −7.70452167699220949313481575430, −7.10739642860083457439317237496, −6.19612950072949939757267547128, −5.07156229072422456816310673147, −3.15576146990859265651399961220, −2.27382941578790942647362525781, −0.62111438818050233261484711067, 1.69806550904465297243744247141, 3.34354675687054925249476296885, 4.33417152981229313593279279001, 5.51302340690430417028363260918, 6.44966189047073980372680076053, 7.80919820650948512233436116547, 8.596076132904021843047446335959, 9.281208635566464104122007587449, 9.973118112397057509966944081811, 10.91823436509602004249527431176

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