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Properties

Label 2-51-51.38-c2-0-8
Degree $2$
Conductor $51$
Sign $0.869 + 0.493i$
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  + 2.52·2-s + (−0.805 − 2.88i)3-s + 2.37·4-s + (0.0213 + 0.0213i)5-s + (−2.03 − 7.29i)6-s + (6.89 + 6.89i)7-s − 4.09·8-s + (−7.70 + 4.65i)9-s + (0.0540 + 0.0540i)10-s + (−2.99 + 2.99i)11-s + (−1.91 − 6.87i)12-s − 5.86·13-s + (17.4 + 17.4i)14-s + (0.0446 − 0.0790i)15-s − 19.8·16-s + (16.8 − 2.04i)17-s + ⋯
L(s)  = 1  + 1.26·2-s + (−0.268 − 0.963i)3-s + 0.594·4-s + (0.00427 + 0.00427i)5-s + (−0.338 − 1.21i)6-s + (0.985 + 0.985i)7-s − 0.511·8-s + (−0.855 + 0.517i)9-s + (0.00540 + 0.00540i)10-s + (−0.272 + 0.272i)11-s + (−0.159 − 0.572i)12-s − 0.451·13-s + (1.24 + 1.24i)14-s + (0.00297 − 0.00527i)15-s − 1.24·16-s + (0.992 − 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.869 + 0.493i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ 0.869 + 0.493i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73613 - 0.457873i\)
\(L(\frac12)\) \(\approx\) \(1.73613 - 0.457873i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.805 + 2.88i)T \)
17 \( 1 + (-16.8 + 2.04i)T \)
good2 \( 1 - 2.52T + 4T^{2} \)
5 \( 1 + (-0.0213 - 0.0213i)T + 25iT^{2} \)
7 \( 1 + (-6.89 - 6.89i)T + 49iT^{2} \)
11 \( 1 + (2.99 - 2.99i)T - 121iT^{2} \)
13 \( 1 + 5.86T + 169T^{2} \)
19 \( 1 + 22.1iT - 361T^{2} \)
23 \( 1 + (4.33 - 4.33i)T - 529iT^{2} \)
29 \( 1 + (-2.07 - 2.07i)T + 841iT^{2} \)
31 \( 1 + (35.3 - 35.3i)T - 961iT^{2} \)
37 \( 1 + (-33.3 + 33.3i)T - 1.36e3iT^{2} \)
41 \( 1 + (-44.4 + 44.4i)T - 1.68e3iT^{2} \)
43 \( 1 + 19.1iT - 1.84e3T^{2} \)
47 \( 1 - 79.9iT - 2.20e3T^{2} \)
53 \( 1 + 54.2T + 2.80e3T^{2} \)
59 \( 1 - 5.34T + 3.48e3T^{2} \)
61 \( 1 + (-44.0 - 44.0i)T + 3.72e3iT^{2} \)
67 \( 1 + 105.T + 4.48e3T^{2} \)
71 \( 1 + (-33.6 - 33.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (20.6 - 20.6i)T - 5.32e3iT^{2} \)
79 \( 1 + (17.7 + 17.7i)T + 6.24e3iT^{2} \)
83 \( 1 - 86.0T + 6.88e3T^{2} \)
89 \( 1 - 87.8iT - 7.92e3T^{2} \)
97 \( 1 + (-66.3 + 66.3i)T - 9.40e3iT^{2} \)
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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

−14.66841096691873385050614918984, −14.10913201662343332469837946545, −12.73779514810839212700465415441, −12.16938282390929469609823900742, −11.15053788768110086711133536969, −8.917699613802840851301460153687, −7.46908301775449932290660790071, −5.86027199102656255771101699214, −4.91807852425720733246080296183, −2.51244201856209923316347503053, 3.59500975303000650118074296160, 4.72763732362370038806786776821, 5.83809176189052225788018959512, 7.88792826784850141018445744988, 9.686486466364852417171202833341, 10.94695757833709233790681908737, 11.90929597458660061478229518439, 13.28867409972293700851177399613, 14.54733322955754588056491778884, 14.78254295430305112615209473031

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