Properties

Label 2-570-57.56-c1-0-3
Degree $2$
Conductor $570$
Sign $0.253 - 0.967i$
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 + (−1.67 − 0.423i)3-s + 4-s + i·5-s + (−1.67 − 0.423i)6-s − 2.47·7-s + 8-s + (2.64 + 1.42i)9-s + i·10-s + 2i·11-s + (−1.67 − 0.423i)12-s + 1.15i·13-s − 2.47·14-s + (0.423 − 1.67i)15-s + 16-s + 6.67i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.969 − 0.244i)3-s + 0.5·4-s + 0.447i·5-s + (−0.685 − 0.173i)6-s − 0.934·7-s + 0.353·8-s + (0.880 + 0.474i)9-s + 0.316i·10-s + 0.603i·11-s + (−0.484 − 0.122i)12-s + 0.319i·13-s − 0.660·14-s + (0.109 − 0.433i)15-s + 0.250·16-s + 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.253 - 0.967i$
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: \((2,\ 570,\ (\ :1/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04048 + 0.803032i\)
\(L(\frac12)\) \(\approx\) \(1.04048 + 0.803032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.423i)T \)
5 \( 1 - iT \)
19 \( 1 + (-4.35 - 0.0388i)T \)
good7 \( 1 + 2.47T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 1.15iT - 13T^{2} \)
17 \( 1 - 6.67iT - 17T^{2} \)
23 \( 1 - 5.79iT - 23T^{2} \)
29 \( 1 + 3.35T + 29T^{2} \)
31 \( 1 + 4.80iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 - 7.83T + 41T^{2} \)
43 \( 1 + 0.978T + 43T^{2} \)
47 \( 1 + 3.02iT - 47T^{2} \)
53 \( 1 - 2.33T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 5.75iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 6.94iT - 97T^{2} \)
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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

−11.15228646240616272409950074580, −10.14295071813432284128252387452, −9.583945568990443122885005579190, −7.86799882297431554592093231524, −7.03651509535782925320400516677, −6.24923638935148400568156792013, −5.58094114090127515657353920110, −4.34351932684991148888378980798, −3.32951930052659455684728004466, −1.72909132828000856978625719889, 0.68321368082771928698069066745, 2.86169804252202260323718457170, 3.99615310095181805249574398042, 5.11115050798851559239319262616, 5.74482728870097784554621279933, 6.71382535436016097414201439506, 7.53070806398092478785569219503, 9.068760370269626693718312067199, 9.769509726449168326464905158652, 10.77950656023800204584367987937

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