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Properties

Label 2-51-3.2-c6-0-11
Degree $2$
Conductor $51$
Sign $-0.244 - 0.969i$
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  + 4.81i·2-s + (26.1 − 6.60i)3-s + 40.8·4-s + 186. i·5-s + (31.7 + 126. i)6-s − 455.·7-s + 504. i·8-s + (641. − 345. i)9-s − 898.·10-s + 546. i·11-s + (1.06e3 − 269. i)12-s + 1.27e3·13-s − 2.19e3i·14-s + (1.23e3 + 4.88e3i)15-s + 182.·16-s + 1.19e3i·17-s + ⋯
L(s)  = 1  + 0.601i·2-s + (0.969 − 0.244i)3-s + 0.637·4-s + 1.49i·5-s + (0.147 + 0.583i)6-s − 1.32·7-s + 0.985i·8-s + (0.880 − 0.474i)9-s − 0.898·10-s + 0.410i·11-s + (0.618 − 0.155i)12-s + 0.579·13-s − 0.799i·14-s + (0.365 + 1.44i)15-s + 0.0446·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.244 - 0.969i$
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: \((2,\ 51,\ (\ :3),\ -0.244 - 0.969i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.55923 + 2.00134i\)
\(L(\frac12)\) \(\approx\) \(1.55923 + 2.00134i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-26.1 + 6.60i)T \)
17 \( 1 - 1.19e3iT \)
good2 \( 1 - 4.81iT - 64T^{2} \)
5 \( 1 - 186. iT - 1.56e4T^{2} \)
7 \( 1 + 455.T + 1.17e5T^{2} \)
11 \( 1 - 546. iT - 1.77e6T^{2} \)
13 \( 1 - 1.27e3T + 4.82e6T^{2} \)
19 \( 1 + 3.93e3T + 4.70e7T^{2} \)
23 \( 1 + 6.46e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.82e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.52e4T + 8.87e8T^{2} \)
37 \( 1 - 8.61e4T + 2.56e9T^{2} \)
41 \( 1 + 9.39e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.11e4T + 6.32e9T^{2} \)
47 \( 1 + 1.26e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.46e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.23e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.32e5T + 5.15e10T^{2} \)
67 \( 1 + 5.38e4T + 9.04e10T^{2} \)
71 \( 1 + 4.49e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.16e5T + 1.51e11T^{2} \)
79 \( 1 - 2.93e5T + 2.43e11T^{2} \)
83 \( 1 - 9.52e5iT - 3.26e11T^{2} \)
89 \( 1 - 8.43e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.48e5T + 8.32e11T^{2} \)
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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

−14.80452442044154568075761421621, −13.76487054948613268983316264356, −12.48872019934906536554046562654, −10.86320612448081573244833058056, −9.829473185565949932041098512959, −8.193373939407292895069652199637, −6.81734870034233963344942754318, −6.47798684330111718473064819244, −3.42200092687094909758318284226, −2.39230367534542612278756057097, 1.04367145489773406587821239285, 2.80686784469262550704765074532, 4.16618708865636947768674131306, 6.28339848403332472133080774381, 8.057367188106144065715171570476, 9.274829143502561774105905758755, 10.08410696397077356826589322862, 11.74544016452227187369223631021, 12.99732752395577691972893933074, 13.36859411893640137465114638332

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