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Properties

Label 2-578-17.13-c1-0-13
Degree $2$
Conductor $578$
Sign $0.933 + 0.359i$
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  + i·2-s + (−0.541 − 0.541i)3-s − 4-s + (1.08 + 1.08i)5-s + (0.541 − 0.541i)6-s + (2.61 − 2.61i)7-s − i·8-s − 2.41i·9-s + (−1.08 + 1.08i)10-s + (0.224 − 0.224i)11-s + (0.541 + 0.541i)12-s − 4.82·13-s + (2.61 + 2.61i)14-s − 1.17i·15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.312 − 0.312i)3-s − 0.5·4-s + (0.484 + 0.484i)5-s + (0.220 − 0.220i)6-s + (0.987 − 0.987i)7-s − 0.353i·8-s − 0.804i·9-s + (−0.342 + 0.342i)10-s + (0.0675 − 0.0675i)11-s + (0.156 + 0.156i)12-s − 1.33·13-s + (0.698 + 0.698i)14-s − 0.302i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.933 + 0.359i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ 0.933 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34594 - 0.250395i\)
\(L(\frac12)\) \(\approx\) \(1.34594 - 0.250395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
17 \( 1 \)
good3 \( 1 + (0.541 + 0.541i)T + 3iT^{2} \)
5 \( 1 + (-1.08 - 1.08i)T + 5iT^{2} \)
7 \( 1 + (-2.61 + 2.61i)T - 7iT^{2} \)
11 \( 1 + (-0.224 + 0.224i)T - 11iT^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (-5.22 + 5.22i)T - 23iT^{2} \)
29 \( 1 + (-2.61 - 2.61i)T + 29iT^{2} \)
31 \( 1 + (-1.08 - 1.08i)T + 31iT^{2} \)
37 \( 1 + (-3.69 - 3.69i)T + 37iT^{2} \)
41 \( 1 + (-4.36 + 4.36i)T - 41iT^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 + 4.82iT - 53T^{2} \)
59 \( 1 - 3.41iT - 59T^{2} \)
61 \( 1 + (-2.16 + 2.16i)T - 61iT^{2} \)
67 \( 1 - 4.58T + 67T^{2} \)
71 \( 1 + (-3.69 - 3.69i)T + 71iT^{2} \)
73 \( 1 + (9.14 + 9.14i)T + 73iT^{2} \)
79 \( 1 + (1.53 - 1.53i)T - 79iT^{2} \)
83 \( 1 - 2.24iT - 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 + (1.62 + 1.62i)T + 97iT^{2} \)
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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

−10.63315094614417218409711409439, −9.776122008724742730940522477570, −8.836300420749272992770854540089, −7.73621346317401908678749229630, −6.94150870854483356287185080925, −6.44127846815957007444567695315, −5.08258859163744211796961767779, −4.38263602837411115023603916608, −2.73384983135217796313615894983, −0.872147252041630489137544416670, 1.65758421319128283632202434733, 2.59395031308599468614155035881, 4.31873201274405751916338872042, 5.26074187064389411920636234816, 5.59483121826263120586591239773, 7.45019328911837160293334433962, 8.278009379458722951326299736816, 9.265278628323271471841810072897, 9.886185785668234587606541290296, 10.85756482340240140406141384585

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