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Properties

Label 2-51-51.47-c2-0-7
Degree $2$
Conductor $51$
Sign $0.219 + 0.975i$
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  + 0.208·2-s + (−0.709 − 2.91i)3-s − 3.95·4-s + (5.25 − 5.25i)5-s + (−0.147 − 0.607i)6-s + (2.40 − 2.40i)7-s − 1.65·8-s + (−7.99 + 4.13i)9-s + (1.09 − 1.09i)10-s + (11.4 + 11.4i)11-s + (2.80 + 11.5i)12-s − 8.12·13-s + (0.500 − 0.500i)14-s + (−19.0 − 11.5i)15-s + 15.4·16-s + (13.5 − 10.2i)17-s + ⋯
L(s)  = 1  + 0.104·2-s + (−0.236 − 0.971i)3-s − 0.989·4-s + (1.05 − 1.05i)5-s + (−0.0246 − 0.101i)6-s + (0.343 − 0.343i)7-s − 0.207·8-s + (−0.888 + 0.459i)9-s + (0.109 − 0.109i)10-s + (1.04 + 1.04i)11-s + (0.234 + 0.961i)12-s − 0.624·13-s + (0.0357 − 0.0357i)14-s + (−1.27 − 0.772i)15-s + 0.967·16-s + (0.798 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.219 + 0.975i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ 0.219 + 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.839302 - 0.671310i\)
\(L(\frac12)\) \(\approx\) \(0.839302 - 0.671310i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.709 + 2.91i)T \)
17 \( 1 + (-13.5 + 10.2i)T \)
good2 \( 1 - 0.208T + 4T^{2} \)
5 \( 1 + (-5.25 + 5.25i)T - 25iT^{2} \)
7 \( 1 + (-2.40 + 2.40i)T - 49iT^{2} \)
11 \( 1 + (-11.4 - 11.4i)T + 121iT^{2} \)
13 \( 1 + 8.12T + 169T^{2} \)
19 \( 1 - 24.0iT - 361T^{2} \)
23 \( 1 + (5.74 + 5.74i)T + 529iT^{2} \)
29 \( 1 + (-11.7 + 11.7i)T - 841iT^{2} \)
31 \( 1 + (-5.33 - 5.33i)T + 961iT^{2} \)
37 \( 1 + (42.7 + 42.7i)T + 1.36e3iT^{2} \)
41 \( 1 + (-8.11 - 8.11i)T + 1.68e3iT^{2} \)
43 \( 1 + 2.45iT - 1.84e3T^{2} \)
47 \( 1 - 64.4iT - 2.20e3T^{2} \)
53 \( 1 + 9.84T + 2.80e3T^{2} \)
59 \( 1 + 73.8T + 3.48e3T^{2} \)
61 \( 1 + (55.3 - 55.3i)T - 3.72e3iT^{2} \)
67 \( 1 - 27.1T + 4.48e3T^{2} \)
71 \( 1 + (-6.71 + 6.71i)T - 5.04e3iT^{2} \)
73 \( 1 + (1.60 + 1.60i)T + 5.32e3iT^{2} \)
79 \( 1 + (64.8 - 64.8i)T - 6.24e3iT^{2} \)
83 \( 1 - 64.2T + 6.88e3T^{2} \)
89 \( 1 + 142. iT - 7.92e3T^{2} \)
97 \( 1 + (-48.5 - 48.5i)T + 9.40e3iT^{2} \)
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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

−14.31659009692429489991987112498, −13.98410165739690527101621197913, −12.60204649729592619311024852254, −12.21792876341968940596616150830, −9.991176568653348631670795018663, −9.028748200578929946960825090241, −7.65675951934727674170945691519, −5.87571656522791921963749852296, −4.67030859985374090410155697482, −1.41639691455380562271269218035, 3.34900612313177160386476749513, 5.13198584716315087720574610542, 6.29719364791889850435899764670, 8.674267812948389024446159422880, 9.648956571179413029974075995075, 10.60031503757748589150849032377, 11.88915619259250264933091229587, 13.71133911503737012489284377724, 14.35130965702606015390814371208, 15.17194938020847697900068060023

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