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Properties

Label 2-57-19.7-c1-0-3
Degree $2$
Conductor $57$
Sign $-0.721 + 0.692i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (−1.04 − 1.80i)2-s + (−0.5 − 0.866i)3-s + (−1.17 + 2.03i)4-s + (−0.675 − 1.17i)5-s + (−1.04 + 1.80i)6-s + 0.351·7-s + 0.734·8-s + (−0.499 + 0.866i)9-s + (−1.41 + 2.44i)10-s + 5.52·11-s + 2.35·12-s + (2.58 − 4.47i)13-s + (−0.367 − 0.635i)14-s + (−0.675 + 1.17i)15-s + (1.58 + 2.74i)16-s + ⋯
L(s)  = 1  + (−0.737 − 1.27i)2-s + (−0.288 − 0.499i)3-s + (−0.587 + 1.01i)4-s + (−0.302 − 0.523i)5-s + (−0.425 + 0.737i)6-s + 0.133·7-s + 0.259·8-s + (−0.166 + 0.288i)9-s + (−0.445 + 0.772i)10-s + 1.66·11-s + 0.678·12-s + (0.717 − 1.24i)13-s + (−0.0981 − 0.169i)14-s + (−0.174 + 0.302i)15-s + (0.396 + 0.686i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.721 + 0.692i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ -0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209183 - 0.520428i\)
\(L(\frac12)\) \(\approx\) \(0.209183 - 0.520428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (2.43 - 3.61i)T \)
good2 \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.675 + 1.17i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.351T + 7T^{2} \)
11 \( 1 - 5.52T + 11T^{2} \)
13 \( 1 + (-2.58 + 4.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.41 - 7.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.35 - 2.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.524T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.35 - 2.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.26 - 5.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.02 + 3.51i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.76 - 4.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.938 + 1.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.99 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.52 + 4.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.85 + 6.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.91 + 6.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.34T + 83T^{2} \)
89 \( 1 + (-2.32 + 4.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.90 - 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.70826039606738999426689963433, −13.17967328770234491298212002195, −12.17509903695284885953826405022, −11.51030442507292996928342453109, −10.32137139330550745859504170295, −9.032811416233504471939533376054, −8.002195579321495856740009216699, −6.03785838530267181462580340344, −3.72131723958425200607757363456, −1.36896615447309439343496612575, 4.17700190894317738548401858803, 6.23236419385355723520457615136, 6.94008622201490174816918985161, 8.618678728878695840298435504181, 9.386965400084828628150560135319, 10.95113834603095607726545469876, 12.00794958446947199983094699605, 14.14405732213462761781038565335, 14.78147888604863489333849665197, 15.85185809122776284486672892594

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