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Properties

Label 2-578-17.16-c1-0-21
Degree $2$
Conductor $578$
Sign $-0.743 + 0.669i$
Analytic cond. $4.61535$
Root an. cond. $2.14833$
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 − 3.22i·3-s + 4-s − 1.18i·5-s − 3.22i·6-s + 1.12i·7-s + 8-s − 7.41·9-s − 1.18i·10-s − 3.41i·11-s − 3.22i·12-s + 0.347·13-s + 1.12i·14-s − 3.82·15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.86i·3-s + 0.5·4-s − 0.529i·5-s − 1.31i·6-s + 0.423i·7-s + 0.353·8-s − 2.47·9-s − 0.374i·10-s − 1.02i·11-s − 0.931i·12-s + 0.0963·13-s + 0.299i·14-s − 0.987·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $-0.743 + 0.669i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ -0.743 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.732337 - 1.90792i\)
\(L(\frac12)\) \(\approx\) \(0.732337 - 1.90792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
17 \( 1 \)
good3 \( 1 + 3.22iT - 3T^{2} \)
5 \( 1 + 1.18iT - 5T^{2} \)
7 \( 1 - 1.12iT - 7T^{2} \)
11 \( 1 + 3.41iT - 11T^{2} \)
13 \( 1 - 0.347T + 13T^{2} \)
19 \( 1 + 0.347T + 19T^{2} \)
23 \( 1 + 0.411iT - 23T^{2} \)
29 \( 1 - 8.78iT - 29T^{2} \)
31 \( 1 + 8.71iT - 31T^{2} \)
37 \( 1 - 0.475iT - 37T^{2} \)
41 \( 1 - 2.63iT - 41T^{2} \)
43 \( 1 - 9.33T + 43T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + 8.41T + 53T^{2} \)
59 \( 1 - 6.41T + 59T^{2} \)
61 \( 1 + 5.70iT - 61T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + 7.59iT - 71T^{2} \)
73 \( 1 + 9.04iT - 73T^{2} \)
79 \( 1 - 13.2iT - 79T^{2} \)
83 \( 1 - 7.73T + 83T^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 + 0.221iT - 97T^{2} \)
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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

−10.92569760123174993974916353337, −9.148823742973365308142201943675, −8.367735723918052008517141501289, −7.62627563509230935489521643842, −6.59394893239411087852863591491, −5.95468279295811697035706192396, −5.07187724130306598289861474281, −3.32947662109683356473059819566, −2.24405955018228085743435525766, −0.955161983067905960295587158454, 2.59157842905336202035554313217, 3.65512062821720237021217097759, 4.42096646409646140114898307868, 5.17187357327942890259138505147, 6.27078902021522637718394317537, 7.36730864356311002560177157428, 8.613419812790325958226571661532, 9.655171674412958577200425128170, 10.27128215604895773340815762732, 10.89039896543399825590279154421

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