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Properties

Label 2-578-17.13-c1-0-10
Degree $2$
Conductor $578$
Sign $0.115 + 0.993i$
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 + (−2.28 − 2.28i)3-s − 4-s + (0.837 + 0.837i)5-s + (2.28 − 2.28i)6-s + (0.792 − 0.792i)7-s − i·8-s + 7.41i·9-s + (−0.837 + 0.837i)10-s + (2.41 − 2.41i)11-s + (2.28 + 2.28i)12-s − 0.347·13-s + (0.792 + 0.792i)14-s − 3.82i·15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−1.31 − 1.31i)3-s − 0.5·4-s + (0.374 + 0.374i)5-s + (0.931 − 0.931i)6-s + (0.299 − 0.299i)7-s − 0.353i·8-s + 2.47i·9-s + (−0.264 + 0.264i)10-s + (0.727 − 0.727i)11-s + (0.658 + 0.658i)12-s − 0.0963·13-s + (0.211 + 0.211i)14-s − 0.987i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.115 + 0.993i$
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: \((2,\ 578,\ (\ :1/2),\ 0.115 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.587873 - 0.523251i\)
\(L(\frac12)\) \(\approx\) \(0.587873 - 0.523251i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.28 + 2.28i)T + 3iT^{2} \)
5 \( 1 + (-0.837 - 0.837i)T + 5iT^{2} \)
7 \( 1 + (-0.792 + 0.792i)T - 7iT^{2} \)
11 \( 1 + (-2.41 + 2.41i)T - 11iT^{2} \)
13 \( 1 + 0.347T + 13T^{2} \)
19 \( 1 + 0.347iT - 19T^{2} \)
23 \( 1 + (-0.290 + 0.290i)T - 23iT^{2} \)
29 \( 1 + (6.20 + 6.20i)T + 29iT^{2} \)
31 \( 1 + (6.16 + 6.16i)T + 31iT^{2} \)
37 \( 1 + (-0.336 - 0.336i)T + 37iT^{2} \)
41 \( 1 + (-1.86 + 1.86i)T - 41iT^{2} \)
43 \( 1 + 9.33iT - 43T^{2} \)
47 \( 1 - 7.86T + 47T^{2} \)
53 \( 1 + 8.41iT - 53T^{2} \)
59 \( 1 + 6.41iT - 59T^{2} \)
61 \( 1 + (4.03 - 4.03i)T - 61iT^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + (5.37 + 5.37i)T + 71iT^{2} \)
73 \( 1 + (-6.39 - 6.39i)T + 73iT^{2} \)
79 \( 1 + (9.38 - 9.38i)T - 79iT^{2} \)
83 \( 1 - 7.73iT - 83T^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 + (-0.156 - 0.156i)T + 97iT^{2} \)
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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

−10.83463890085990575767115445117, −9.659733201241992432801619670587, −8.393688243799968318241255323044, −7.48845795334404127268529064543, −6.85383081715829743239986241247, −5.98069857193544472203757492628, −5.52631969444135299247518577700, −4.13255183002527393021810649152, −2.06476055091916733400170761246, −0.56700663256554985126282221056, 1.47181427342796815824098840057, 3.45556013243580804884489637973, 4.45099688527098419068603977886, 5.18801842852323745489979005030, 5.91582655175101720162751828726, 7.18509141852802119522334073162, 9.019110264005313532143960226942, 9.279353370608736929739655631858, 10.20592046950514489391152503357, 10.93082116207837149335952328989

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