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Properties

Label 2-51-3.2-c6-0-1
Degree $2$
Conductor $51$
Sign $-0.967 - 0.254i$
Analytic cond. $11.7327$
Root an. cond. $3.42531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.35i·2-s + (−6.87 + 26.1i)3-s + 23.6·4-s + 214. i·5-s + (165. + 43.6i)6-s − 487.·7-s − 556. i·8-s + (−634. − 359. i)9-s + 1.36e3·10-s − 1.78e3i·11-s + (−162. + 617. i)12-s − 1.81e3·13-s + 3.09e3i·14-s + (−5.59e3 − 1.47e3i)15-s − 2.02e3·16-s − 1.19e3i·17-s + ⋯
L(s)  = 1  − 0.793i·2-s + (−0.254 + 0.967i)3-s + 0.369·4-s + 1.71i·5-s + (0.767 + 0.202i)6-s − 1.42·7-s − 1.08i·8-s + (−0.870 − 0.492i)9-s + 1.36·10-s − 1.34i·11-s + (−0.0941 + 0.357i)12-s − 0.826·13-s + 1.12i·14-s + (−1.65 − 0.436i)15-s − 0.493·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.967 - 0.254i$
Analytic conductor: \(11.7327\)
Root analytic conductor: \(3.42531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :3),\ -0.967 - 0.254i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0504149 + 0.389437i\)
\(L(\frac12)\) \(\approx\) \(0.0504149 + 0.389437i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.87 - 26.1i)T \)
17 \( 1 + 1.19e3iT \)
good2 \( 1 + 6.35iT - 64T^{2} \)
5 \( 1 - 214. iT - 1.56e4T^{2} \)
7 \( 1 + 487.T + 1.17e5T^{2} \)
11 \( 1 + 1.78e3iT - 1.77e6T^{2} \)
13 \( 1 + 1.81e3T + 4.82e6T^{2} \)
19 \( 1 - 152.T + 4.70e7T^{2} \)
23 \( 1 - 1.21e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.89e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.34e3T + 8.87e8T^{2} \)
37 \( 1 + 3.63e4T + 2.56e9T^{2} \)
41 \( 1 - 4.62e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.74e4T + 6.32e9T^{2} \)
47 \( 1 - 3.74e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.37e5iT - 2.21e10T^{2} \)
59 \( 1 - 9.98e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.13e5T + 5.15e10T^{2} \)
67 \( 1 + 3.74e5T + 9.04e10T^{2} \)
71 \( 1 + 1.78e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.89e5T + 1.51e11T^{2} \)
79 \( 1 - 7.69e5T + 2.43e11T^{2} \)
83 \( 1 - 1.55e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.85e4iT - 4.96e11T^{2} \)
97 \( 1 + 1.12e6T + 8.32e11T^{2} \)
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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

−14.91790525232465088689089189319, −13.66195613030121055845291312338, −12.02552691679070941134333063881, −11.01697641862528501101542658050, −10.34316072428526858214294080207, −9.453249647731489723975787969886, −7.01049000895856501073858282807, −6.03138360771578754340628397121, −3.33101828299059781380914978057, −3.03858527465114391623328321466, 0.16070746911115146690506169201, 2.08990032157680625355255529945, 4.93263382003067461738192937209, 6.22932998751734118266537371241, 7.30305764103437340529694906428, 8.500080510224066094980143588764, 9.865941189941337834223270531473, 12.04095978262761711511524746583, 12.51281541853282384712787624443, 13.48502985732219562228079110590

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