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Properties

Label 2-570-5.4-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.241 - 0.970i$
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  + i·2-s + i·3-s − 4-s + (−0.539 − 2.17i)5-s − 6-s + 1.07i·7-s − i·8-s − 9-s + (2.17 − 0.539i)10-s + 6.34·11-s − i·12-s + 3.41i·13-s − 1.07·14-s + (2.17 − 0.539i)15-s + 16-s + 5.41i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.241 − 0.970i)5-s − 0.408·6-s + 0.407i·7-s − 0.353i·8-s − 0.333·9-s + (0.686 − 0.170i)10-s + 1.91·11-s − 0.288i·12-s + 0.948i·13-s − 0.288·14-s + (0.560 − 0.139i)15-s + 0.250·16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.241 - 0.970i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833140 + 1.06547i\)
\(L(\frac12)\) \(\approx\) \(0.833140 + 1.06547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + (0.539 + 2.17i)T \)
19 \( 1 + T \)
good7 \( 1 - 1.07iT - 7T^{2} \)
11 \( 1 - 6.34T + 11T^{2} \)
13 \( 1 - 3.41iT - 13T^{2} \)
17 \( 1 - 5.41iT - 17T^{2} \)
23 \( 1 - 6.34iT - 23T^{2} \)
29 \( 1 - 0.340T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 3.41iT - 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 - 6.34iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 0.738T + 59T^{2} \)
61 \( 1 + 2.68T + 61T^{2} \)
67 \( 1 + 2.83iT - 67T^{2} \)
71 \( 1 + 2.83T + 71T^{2} \)
73 \( 1 - 6.83iT - 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 - 0.894iT - 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 - 3.65iT - 97T^{2} \)
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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

−11.06204503847437797458908113920, −9.692532806885252921889493876208, −9.044600509040484211592580696179, −8.651568588817250227679040094782, −7.44340415357525795642276132126, −6.29470379124610929477660213149, −5.56564875337071980002908405314, −4.25119854084832143991695701691, −3.88730176969095619246564633366, −1.53183109142560054556249760847, 0.883583711262000729536427883987, 2.50701061230536849326127293194, 3.49289241883032846454710924905, 4.56421428813082819501461395220, 6.14232154303780458316053637612, 6.85858959030907383185731110403, 7.76205384826580137483446425049, 8.822681838426324220010912001987, 9.767912359554629135645061041020, 10.62029088868600397900563173691

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