Properties

Label 2-570-57.56-c1-0-9
Degree $2$
Conductor $570$
Sign $0.927 + 0.374i$
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 + (−1 + 1.41i)3-s + 4-s i·5-s + (1 − 1.41i)6-s + 3.41·7-s − 8-s + (−1.00 − 2.82i)9-s + i·10-s + 2.58i·11-s + (−1 + 1.41i)12-s − 6.24i·13-s − 3.41·14-s + (1.41 + i)15-s + 16-s − 2.82i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.577 + 0.816i)3-s + 0.5·4-s − 0.447i·5-s + (0.408 − 0.577i)6-s + 1.29·7-s − 0.353·8-s + (−0.333 − 0.942i)9-s + 0.316i·10-s + 0.779i·11-s + (−0.288 + 0.408i)12-s − 1.73i·13-s − 0.912·14-s + (0.365 + 0.258i)15-s + 0.250·16-s − 0.685i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.927 + 0.374i$
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: \((2,\ 570,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903618 - 0.175659i\)
\(L(\frac12)\) \(\approx\) \(0.903618 - 0.175659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.41i)T \)
5 \( 1 + iT \)
19 \( 1 + (4.24 - i)T \)
good7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
13 \( 1 + 6.24iT - 13T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 9.07T + 29T^{2} \)
31 \( 1 - 1.17iT - 31T^{2} \)
37 \( 1 + 6.24iT - 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 - 9.41T + 43T^{2} \)
47 \( 1 - 8.82iT - 47T^{2} \)
53 \( 1 - 8.82T + 53T^{2} \)
59 \( 1 - 7.17T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 9.31iT - 79T^{2} \)
83 \( 1 + 7.17iT - 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 9.41iT - 97T^{2} \)
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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

−10.46806142761083857396304238053, −10.10049205314492014644531484981, −8.823480936306068995985327888705, −8.297655785930812812975445660672, −7.27667189790026424773862825380, −5.98540694012270788031111082900, −5.03887163466081568366153761917, −4.30541185027600792492673443088, −2.58225121220856977310922473630, −0.804001058421859420290469090962, 1.35657485493794321942914727266, 2.33839329309589272615354702397, 4.22122000047918702823790887336, 5.50175612034016594296986682554, 6.52713972028414602226997519436, 7.14946053955124206373481033098, 8.277263154047694248599380089730, 8.658180644461699747566894654653, 10.10605250084482035912886282758, 10.99044168036423299835057418374

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