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Properties

Label 2-572-572.571-c1-0-79
Degree $2$
Conductor $572$
Sign $-0.981 - 0.193i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + (1.09 − 0.897i)2-s − 3.43i·3-s + (0.387 − 1.96i)4-s + (−3.08 − 3.74i)6-s + 0.803i·7-s + (−1.33 − 2.49i)8-s − 8.76·9-s + 3.31i·11-s + (−6.73 − 1.33i)12-s + 3.60·13-s + (0.721 + 0.878i)14-s + (−3.69 − 1.52i)16-s + (−9.58 + 7.87i)18-s − 5.18i·19-s + 2.75·21-s + (2.97 + 3.62i)22-s + ⋯
L(s)  = 1  + (0.772 − 0.634i)2-s − 1.98i·3-s + (0.193 − 0.981i)4-s + (−1.25 − 1.53i)6-s + 0.303i·7-s + (−0.472 − 0.881i)8-s − 2.92·9-s + 1.00i·11-s + (−1.94 − 0.384i)12-s + 1.00·13-s + (0.192 + 0.234i)14-s + (−0.924 − 0.380i)16-s + (−2.25 + 1.85i)18-s − 1.18i·19-s + 0.601·21-s + (0.634 + 0.772i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.981 - 0.193i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.981 - 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194717 + 1.98922i\)
\(L(\frac12)\) \(\approx\) \(0.194717 + 1.98922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.897i)T \)
11 \( 1 - 3.31iT \)
13 \( 1 - 3.60T \)
good3 \( 1 + 3.43iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 0.803iT - 7T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.18iT - 19T^{2} \)
23 \( 1 + 6.19iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.3T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 17.5iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.80886940694663679130799044596, −9.261861733990474208217858596839, −8.456801298358837846374828523055, −7.24408241764389975458628815257, −6.59116676180800172229585646291, −5.84100954509966143808477719880, −4.65967714336773319646402653537, −2.95250624924913875469115979637, −2.13937067100749525007343031336, −0.921942173633447648974562236669, 3.14096680827122388999383302054, 3.69913496058802879251984971655, 4.58175797683310980747051492451, 5.65226501947713997702682529565, 6.11099088458276593567742813932, 7.80697375369155062452627481371, 8.646193917520954308679407923442, 9.322855243735964303742587772134, 10.49868362107688148289099329935, 11.09681498017164047822026037790

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