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Properties

Label 2-51-51.50-c2-0-7
Degree $2$
Conductor $51$
Sign $-0.472 + 0.881i$
Analytic cond. $1.38964$
Root an. cond. $1.17883$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.84i·2-s + (−0.901 − 2.86i)3-s + 0.582·4-s − 2.44·5-s + (−5.28 + 1.66i)6-s + 1.19i·7-s − 8.47i·8-s + (−7.37 + 5.16i)9-s + 4.52i·10-s + 12.8·11-s + (−0.525 − 1.66i)12-s + 11.1·13-s + 2.20·14-s + (2.20 + 7.00i)15-s − 13.3·16-s + (11.8 + 12.1i)17-s + ⋯
L(s)  = 1  − 0.924i·2-s + (−0.300 − 0.953i)3-s + 0.145·4-s − 0.489·5-s + (−0.881 + 0.277i)6-s + 0.170i·7-s − 1.05i·8-s + (−0.819 + 0.573i)9-s + 0.452i·10-s + 1.17·11-s + (−0.0437 − 0.138i)12-s + 0.858·13-s + 0.157·14-s + (0.147 + 0.467i)15-s − 0.833·16-s + (0.698 + 0.715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.472 + 0.881i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ -0.472 + 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.567164 - 0.947996i\)
\(L(\frac12)\) \(\approx\) \(0.567164 - 0.947996i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.901 + 2.86i)T \)
17 \( 1 + (-11.8 - 12.1i)T \)
good2 \( 1 + 1.84iT - 4T^{2} \)
5 \( 1 + 2.44T + 25T^{2} \)
7 \( 1 - 1.19iT - 49T^{2} \)
11 \( 1 - 12.8T + 121T^{2} \)
13 \( 1 - 11.1T + 169T^{2} \)
19 \( 1 + 2.74T + 361T^{2} \)
23 \( 1 - 14.0T + 529T^{2} \)
29 \( 1 + 38.5T + 841T^{2} \)
31 \( 1 - 44.5iT - 961T^{2} \)
37 \( 1 + 34.0iT - 1.36e3T^{2} \)
41 \( 1 + 58.0T + 1.68e3T^{2} \)
43 \( 1 + 1.16T + 1.84e3T^{2} \)
47 \( 1 - 56.8iT - 2.20e3T^{2} \)
53 \( 1 + 19.2iT - 2.80e3T^{2} \)
59 \( 1 - 39.1iT - 3.48e3T^{2} \)
61 \( 1 - 86.5iT - 3.72e3T^{2} \)
67 \( 1 - 55.5T + 4.48e3T^{2} \)
71 \( 1 + 25.4T + 5.04e3T^{2} \)
73 \( 1 + 119. iT - 5.32e3T^{2} \)
79 \( 1 + 96.5iT - 6.24e3T^{2} \)
83 \( 1 + 33.2iT - 6.88e3T^{2} \)
89 \( 1 + 168. iT - 7.92e3T^{2} \)
97 \( 1 - 29.5iT - 9.40e3T^{2} \)
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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

−14.74684447836089049260555088213, −13.34856785480937248564552158256, −12.29294627964744611370050241687, −11.61401016755427468374739848643, −10.64293501069519268680507649529, −8.879576454944746748300777563203, −7.33089936623867053031050888675, −6.09781472550798687940720749481, −3.58181168726986202142985212459, −1.49649850718889595580400374123, 3.80329767339686070086888289863, 5.49773853157214704726905658884, 6.77335490906774580941108126294, 8.263505766343986672335482317384, 9.557699704839095582857157647604, 11.16520841902078744489910517659, 11.78254851334308717666357380664, 13.85409491124803491712423520502, 14.96958192174850639952935977039, 15.58043046403722842809110675460

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