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Properties

Label 2-570-95.37-c1-0-4
Degree $2$
Conductor $570$
Sign $0.953 - 0.300i$
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  + (0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (−1.66 + 1.49i)5-s + 1.00·6-s + (−0.170 − 0.170i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−0.120 + 2.23i)10-s + 3.43·11-s + (0.707 − 0.707i)12-s + (4.54 + 4.54i)13-s − 0.240·14-s + (−2.23 − 0.120i)15-s − 1.00·16-s + (0.537 + 0.537i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (−0.744 + 0.668i)5-s + 0.408·6-s + (−0.0643 − 0.0643i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.0380 + 0.706i)10-s + 1.03·11-s + (0.204 − 0.204i)12-s + (1.25 + 1.25i)13-s − 0.0643·14-s + (−0.576 − 0.0310i)15-s − 0.250·16-s + (0.130 + 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.953 - 0.300i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01371 + 0.309567i\)
\(L(\frac12)\) \(\approx\) \(2.01371 + 0.309567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.66 - 1.49i)T \)
19 \( 1 + (-2.42 - 3.62i)T \)
good7 \( 1 + (0.170 + 0.170i)T + 7iT^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 + (-4.54 - 4.54i)T + 13iT^{2} \)
17 \( 1 + (-0.537 - 0.537i)T + 17iT^{2} \)
23 \( 1 + (-5.15 + 5.15i)T - 23iT^{2} \)
29 \( 1 + 5.37T + 29T^{2} \)
31 \( 1 - 5.37iT - 31T^{2} \)
37 \( 1 + (5.54 - 5.54i)T - 37iT^{2} \)
41 \( 1 - 3.68iT - 41T^{2} \)
43 \( 1 + (-2.29 + 2.29i)T - 43iT^{2} \)
47 \( 1 + (9.57 + 9.57i)T + 47iT^{2} \)
53 \( 1 + (1.93 + 1.93i)T + 53iT^{2} \)
59 \( 1 + 4.20T + 59T^{2} \)
61 \( 1 - 1.65T + 61T^{2} \)
67 \( 1 + (-0.481 + 0.481i)T - 67iT^{2} \)
71 \( 1 + 8.93iT - 71T^{2} \)
73 \( 1 + (-8.74 + 8.74i)T - 73iT^{2} \)
79 \( 1 - 1.15T + 79T^{2} \)
83 \( 1 + (9.97 - 9.97i)T - 83iT^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \)
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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

−10.94000033165450407097597407986, −10.07653083565753964868976481325, −9.054602941505352881899304561766, −8.348469705054110507170681616635, −6.95936735203920946242588996454, −6.35805951548625947401226584001, −4.85125622236662708838737287810, −3.77689652509866896515370062021, −3.35860650366128964006628490459, −1.66383942899470364336610096666, 1.13083234475132818666413665332, 3.19185692605158870178146101315, 3.90725143617547161485136019021, 5.18622620254810849118499768389, 6.09322312351371460821935910187, 7.27659198727543861868487389030, 7.85145889023459127970219561691, 8.859862947519289974498149218963, 9.364372429377285983045401192396, 11.10657617636751806514389038701

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