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Properties

Label 2-570-15.2-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.520 + 0.853i$
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 + (−1.63 + 0.581i)3-s − 1.00i·4-s + (−1.36 + 1.76i)5-s + (0.742 − 1.56i)6-s + (0.102 + 0.102i)7-s + (0.707 + 0.707i)8-s + (2.32 − 1.89i)9-s + (−0.282 − 2.21i)10-s + 6.12i·11-s + (0.581 + 1.63i)12-s + (−0.754 + 0.754i)13-s − 0.145·14-s + (1.20 − 3.68i)15-s − 1.00·16-s + (−2.33 + 2.33i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.941 + 0.335i)3-s − 0.500i·4-s + (−0.612 + 0.790i)5-s + (0.302 − 0.638i)6-s + (0.0388 + 0.0388i)7-s + (0.250 + 0.250i)8-s + (0.774 − 0.632i)9-s + (−0.0893 − 0.701i)10-s + 1.84i·11-s + (0.167 + 0.470i)12-s + (−0.209 + 0.209i)13-s − 0.0388·14-s + (0.310 − 0.950i)15-s − 0.250·16-s + (−0.565 + 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.520 + 0.853i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.520 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0820917 - 0.146273i\)
\(L(\frac12)\) \(\approx\) \(0.0820917 - 0.146273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.581i)T \)
5 \( 1 + (1.36 - 1.76i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.102 - 0.102i)T + 7iT^{2} \)
11 \( 1 - 6.12iT - 11T^{2} \)
13 \( 1 + (0.754 - 0.754i)T - 13iT^{2} \)
17 \( 1 + (2.33 - 2.33i)T - 17iT^{2} \)
23 \( 1 + (6.21 + 6.21i)T + 23iT^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + (3.66 + 3.66i)T + 37iT^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (0.994 - 0.994i)T - 43iT^{2} \)
47 \( 1 + (4.88 - 4.88i)T - 47iT^{2} \)
53 \( 1 + (-2.86 - 2.86i)T + 53iT^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + (-4.37 - 4.37i)T + 67iT^{2} \)
71 \( 1 - 0.311iT - 71T^{2} \)
73 \( 1 + (-0.755 + 0.755i)T - 73iT^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 + (11.1 + 11.1i)T + 83iT^{2} \)
89 \( 1 - 2.30T + 89T^{2} \)
97 \( 1 + (0.595 + 0.595i)T + 97iT^{2} \)
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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

−11.10405642558520362942196568041, −10.29608144381130134503701753955, −9.889806467395555380238171728332, −8.654197621523940795127586398416, −7.49560220000156359953981920848, −6.85396646508766089292342899245, −6.14503179340396879972052258907, −4.75572556907346226276116211463, −4.10646459229610820415115931271, −2.12694231988891692773484121181, 0.13369894736689733194983032009, 1.37196729224859878500597219442, 3.24386539562772306256420878158, 4.46968943520567380401414537094, 5.51630464388503367130736888079, 6.47929469378065140655732088650, 7.78608968552944114407965994702, 8.259865556287241114876501305948, 9.331362895706851705914520229130, 10.31393785293175242114710398470

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