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Properties

Label 2-578-17.4-c1-0-19
Degree $2$
Conductor $578$
Sign $-0.151 + 0.988i$
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 + (1.41 − 1.41i)3-s − 4-s + (1.41 + 1.41i)6-s + (−2.82 − 2.82i)7-s − i·8-s − 1.00i·9-s + (−4.24 − 4.24i)11-s + (−1.41 + 1.41i)12-s − 2·13-s + (2.82 − 2.82i)14-s + 16-s + 1.00·18-s − 4i·19-s − 8.00·21-s + (4.24 − 4.24i)22-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.816 − 0.816i)3-s − 0.5·4-s + (0.577 + 0.577i)6-s + (−1.06 − 1.06i)7-s − 0.353i·8-s − 0.333i·9-s + (−1.27 − 1.27i)11-s + (−0.408 + 0.408i)12-s − 0.554·13-s + (0.755 − 0.755i)14-s + 0.250·16-s + 0.235·18-s − 0.917i·19-s − 1.74·21-s + (0.904 − 0.904i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ -0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710268 - 0.827284i\)
\(L(\frac12)\) \(\approx\) \(0.710268 - 0.827284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.41i)T - 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + (2.82 + 2.82i)T + 7iT^{2} \)
11 \( 1 + (4.24 + 4.24i)T + 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + (-2.82 + 2.82i)T - 31iT^{2} \)
37 \( 1 + (-2.82 + 2.82i)T - 37iT^{2} \)
41 \( 1 + (-4.24 - 4.24i)T + 41iT^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (2.82 + 2.82i)T + 61iT^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 71iT^{2} \)
73 \( 1 + (-1.41 + 1.41i)T - 73iT^{2} \)
79 \( 1 + (5.65 + 5.65i)T + 79iT^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-9.89 + 9.89i)T - 97iT^{2} \)
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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

−10.34021397778703989096532162188, −9.397682050273467464598048063531, −8.473114795398940100091675642906, −7.62873863573260673151336387136, −7.14637155377355554539304081641, −6.16264302083702561668458361399, −5.01902070519739784303223315298, −3.55088370004165817998694660510, −2.64683386420208735349147623632, −0.53070282129183466157909749757, 2.39782064731494503880738504547, 2.92508762724022476591227108172, 4.16518240900301913280902737550, 5.11284438512272867514428403664, 6.29941707747763324660347934193, 7.72670709491563663215781400088, 8.604299210363705363873879879388, 9.562538574596311137043234309226, 9.885624896950080306771177854860, 10.57786440423496178388431453277

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