Properties

Label 2-570-285.272-c1-0-32
Degree $2$
Conductor $570$
Sign $-0.773 + 0.633i$
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  + (0.0871 − 0.996i)2-s + (−1.66 − 0.487i)3-s + (−0.984 − 0.173i)4-s + (2.04 − 0.915i)5-s + (−0.630 + 1.61i)6-s + (−0.956 + 0.256i)7-s + (−0.258 + 0.965i)8-s + (2.52 + 1.61i)9-s + (−0.734 − 2.11i)10-s + (1.68 − 0.974i)11-s + (1.55 + 0.768i)12-s + (2.53 − 5.43i)13-s + (0.171 + 0.974i)14-s + (−3.83 + 0.527i)15-s + (0.939 + 0.342i)16-s + (0.249 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (−0.959 − 0.281i)3-s + (−0.492 − 0.0868i)4-s + (0.912 − 0.409i)5-s + (−0.257 + 0.658i)6-s + (−0.361 + 0.0968i)7-s + (−0.0915 + 0.341i)8-s + (0.841 + 0.539i)9-s + (−0.232 − 0.667i)10-s + (0.508 − 0.293i)11-s + (0.448 + 0.221i)12-s + (0.702 − 1.50i)13-s + (0.0459 + 0.260i)14-s + (−0.990 + 0.136i)15-s + (0.234 + 0.0855i)16-s + (0.0604 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.773 + 0.633i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360754 - 1.00977i\)
\(L(\frac12)\) \(\approx\) \(0.360754 - 1.00977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 + 0.996i)T \)
3 \( 1 + (1.66 + 0.487i)T \)
5 \( 1 + (-2.04 + 0.915i)T \)
19 \( 1 + (4.19 - 1.17i)T \)
good7 \( 1 + (0.956 - 0.256i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.68 + 0.974i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.53 + 5.43i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.249 + 2.84i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (2.28 - 1.59i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (0.831 + 0.697i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.207 + 0.358i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.98 + 6.98i)T + 37iT^{2} \)
41 \( 1 + (-0.231 + 0.635i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.540 + 0.378i)T + (14.7 + 40.4i)T^{2} \)
47 \( 1 + (-9.29 + 0.813i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (7.05 - 4.93i)T + (18.1 - 49.8i)T^{2} \)
59 \( 1 + (0.996 - 0.836i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.19 + 6.76i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (1.15 + 13.2i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (-8.83 + 1.55i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.75 + 2.68i)T + (46.9 - 55.9i)T^{2} \)
79 \( 1 + (3.54 - 9.75i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.81 + 1.82i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-15.9 + 5.79i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (10.1 + 0.885i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.57226065344708810247693817898, −9.752566535739760625222315927524, −8.880753225160349575318004049985, −7.77760154797080572715469984983, −6.36546114044947104914923922896, −5.77850112202991337878654530526, −4.91253033404106526116905266253, −3.56220305077060225516184123968, −2.04219895740551583073683431495, −0.70521770258866344335216912591, 1.71267527312550032436701380707, 3.76879004068407238855788258449, 4.66914864721963104586507257629, 5.86298632078918398523029259012, 6.54968435666486005380926288615, 6.91611126187449717228221244073, 8.579069443496722883482989449568, 9.395379527080087279026004186615, 10.16182949831087738687509040280, 10.94333290117903878092647503831

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