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Properties

Label 2-578-17.15-c1-0-12
Degree $2$
Conductor $578$
Sign $0.0465 - 0.998i$
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  + (0.707 + 0.707i)2-s + (0.707 + 1.70i)3-s + 1.00i·4-s + (3.41 − 1.41i)5-s + (−0.707 + 1.70i)6-s + (−1.41 − 0.585i)7-s + (−0.707 + 0.707i)8-s + (−0.292 + 0.292i)9-s + (3.41 + 1.41i)10-s + (−1.70 + 4.12i)11-s + (−1.70 + 0.707i)12-s + 0.828i·13-s + (−0.585 − 1.41i)14-s + (4.82 + 4.82i)15-s − 1.00·16-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.985i)3-s + 0.500i·4-s + (1.52 − 0.632i)5-s + (−0.288 + 0.696i)6-s + (−0.534 − 0.221i)7-s + (−0.250 + 0.250i)8-s + (−0.0976 + 0.0976i)9-s + (1.07 + 0.447i)10-s + (−0.514 + 1.24i)11-s + (−0.492 + 0.204i)12-s + 0.229i·13-s + (−0.156 − 0.377i)14-s + (1.24 + 1.24i)15-s − 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.0465 - 0.998i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ 0.0465 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85256 + 1.76830i\)
\(L(\frac12)\) \(\approx\) \(1.85256 + 1.76830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (-0.707 - 1.70i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.41 + 1.41i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.41 + 0.585i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.70 - 4.12i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.828iT - 13T^{2} \)
19 \( 1 + (-0.585 - 0.585i)T + 19iT^{2} \)
23 \( 1 + (-1.17 + 2.82i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.41 + 0.585i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.41 - 3.41i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.828 + 2i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (10.3 + 4.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.24 - 1.24i)T - 43iT^{2} \)
47 \( 1 + 9.65iT - 47T^{2} \)
53 \( 1 + (-0.585 - 0.585i)T + 53iT^{2} \)
59 \( 1 + (-0.414 + 0.414i)T - 59iT^{2} \)
61 \( 1 + (6.82 + 2.82i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.41T + 67T^{2} \)
71 \( 1 + (0.828 + 2i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.94 + 2.05i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2 + 4.82i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.41 - 4.41i)T + 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + (-5.12 + 2.12i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.35458642967055418413898585525, −10.06929436104410944221000098382, −9.247063505262549317625853587684, −8.566305071095947076492286509313, −7.12027982330794886749419201609, −6.30812650032886482337706997022, −5.12669867897201520669268474177, −4.64415688326894406948897147289, −3.36015352270241007719110812546, −2.02632639180452980452443297465, 1.41033887745691928526893963846, 2.58921857386286403589818305315, 3.17412466075001357697154187536, 5.12739660385398921467698429228, 6.07309506560458725763023368845, 6.57493761571501184198546017715, 7.75722883747127534617735646721, 8.868430127577849604118926374688, 9.812056471597951801557564064774, 10.49293704876012096842190456346

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