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Properties

Label 2-578-17.13-c1-0-3
Degree $2$
Conductor $578$
Sign $0.854 + 0.519i$
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 + (−2.03 − 2.03i)3-s − 4-s + (1.16 + 1.16i)5-s + (−2.03 + 2.03i)6-s + (−1.70 + 1.70i)7-s + i·8-s + 5.29i·9-s + (1.16 − 1.16i)10-s + (−0.330 + 0.330i)11-s + (2.03 + 2.03i)12-s + 6.22·13-s + (1.70 + 1.70i)14-s − 4.75i·15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.17 − 1.17i)3-s − 0.5·4-s + (0.522 + 0.522i)5-s + (−0.831 + 0.831i)6-s + (−0.644 + 0.644i)7-s + 0.353i·8-s + 1.76i·9-s + (0.369 − 0.369i)10-s + (−0.0997 + 0.0997i)11-s + (0.587 + 0.587i)12-s + 1.72·13-s + (0.455 + 0.455i)14-s − 1.22i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.854 + 0.519i$
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: \((2,\ 578,\ (\ :1/2),\ 0.854 + 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863775 - 0.242217i\)
\(L(\frac12)\) \(\approx\) \(0.863775 - 0.242217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 2.03i)T + 3iT^{2} \)
5 \( 1 + (-1.16 - 1.16i)T + 5iT^{2} \)
7 \( 1 + (1.70 - 1.70i)T - 7iT^{2} \)
11 \( 1 + (0.330 - 0.330i)T - 11iT^{2} \)
13 \( 1 - 6.22T + 13T^{2} \)
19 \( 1 - 5.90iT - 19T^{2} \)
23 \( 1 + (1.35 - 1.35i)T - 23iT^{2} \)
29 \( 1 + (-3.46 - 3.46i)T + 29iT^{2} \)
31 \( 1 + (0.592 + 0.592i)T + 31iT^{2} \)
37 \( 1 + (0.822 + 0.822i)T + 37iT^{2} \)
41 \( 1 + (-2.85 + 2.85i)T - 41iT^{2} \)
43 \( 1 + 2.65iT - 43T^{2} \)
47 \( 1 - 3.98T + 47T^{2} \)
53 \( 1 + 1.49iT - 53T^{2} \)
59 \( 1 - 13.8iT - 59T^{2} \)
61 \( 1 + (-3.11 + 3.11i)T - 61iT^{2} \)
67 \( 1 - 2.10T + 67T^{2} \)
71 \( 1 + (-8.67 - 8.67i)T + 71iT^{2} \)
73 \( 1 + (-9.22 - 9.22i)T + 73iT^{2} \)
79 \( 1 + (-6.23 + 6.23i)T - 79iT^{2} \)
83 \( 1 + 8.88iT - 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + (-9.19 - 9.19i)T + 97iT^{2} \)
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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

−10.74592475976669266326361795186, −10.16949967717068603780962972063, −8.945013287007300563217633620963, −7.959967364427095386818983409207, −6.71709329270663382502311682821, −6.03905235305995197768585064247, −5.50812856656254200296293354216, −3.75414255577437790925032510024, −2.36635915666437849981720276966, −1.21455136634210890830202853727, 0.71148077299371216804222508146, 3.53950478440653801155448063366, 4.44802918646134051045107184266, 5.31731459516113876301759263244, 6.15864560052007439725284595247, 6.75570238943462237922467500778, 8.298132477044276138405083182712, 9.266839113839256157352418999000, 9.819544844165658320577526189025, 10.79824626100593681477347289722

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