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Properties

Label 2-51-51.38-c2-0-0
Degree $2$
Conductor $51$
Sign $0.660 - 0.751i$
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  − 2.55·2-s + (−2.60 − 1.48i)3-s + 2.52·4-s + (5.68 + 5.68i)5-s + (6.66 + 3.78i)6-s + (1.53 + 1.53i)7-s + 3.75·8-s + (4.61 + 7.72i)9-s + (−14.5 − 14.5i)10-s + (−9.63 + 9.63i)11-s + (−6.59 − 3.74i)12-s + 10.8·13-s + (−3.92 − 3.92i)14-s + (−6.41 − 23.2i)15-s − 19.7·16-s + (12.9 − 10.9i)17-s + ⋯
L(s)  = 1  − 1.27·2-s + (−0.869 − 0.493i)3-s + 0.632·4-s + (1.13 + 1.13i)5-s + (1.11 + 0.630i)6-s + (0.219 + 0.219i)7-s + 0.469·8-s + (0.513 + 0.858i)9-s + (−1.45 − 1.45i)10-s + (−0.876 + 0.876i)11-s + (−0.549 − 0.311i)12-s + 0.834·13-s + (−0.280 − 0.280i)14-s + (−0.427 − 1.54i)15-s − 1.23·16-s + (0.762 − 0.646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.660 - 0.751i$
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: \((2,\ 51,\ (\ :1),\ 0.660 - 0.751i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.490465 + 0.221920i\)
\(L(\frac12)\) \(\approx\) \(0.490465 + 0.221920i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.60 + 1.48i)T \)
17 \( 1 + (-12.9 + 10.9i)T \)
good2 \( 1 + 2.55T + 4T^{2} \)
5 \( 1 + (-5.68 - 5.68i)T + 25iT^{2} \)
7 \( 1 + (-1.53 - 1.53i)T + 49iT^{2} \)
11 \( 1 + (9.63 - 9.63i)T - 121iT^{2} \)
13 \( 1 - 10.8T + 169T^{2} \)
19 \( 1 - 22.1iT - 361T^{2} \)
23 \( 1 + (9.91 - 9.91i)T - 529iT^{2} \)
29 \( 1 + (-4.19 - 4.19i)T + 841iT^{2} \)
31 \( 1 + (-13.4 + 13.4i)T - 961iT^{2} \)
37 \( 1 + (-6.27 + 6.27i)T - 1.36e3iT^{2} \)
41 \( 1 + (17.0 - 17.0i)T - 1.68e3iT^{2} \)
43 \( 1 + 30.2iT - 1.84e3T^{2} \)
47 \( 1 + 57.7iT - 2.20e3T^{2} \)
53 \( 1 + 74.4T + 2.80e3T^{2} \)
59 \( 1 - 84.8T + 3.48e3T^{2} \)
61 \( 1 + (23.5 + 23.5i)T + 3.72e3iT^{2} \)
67 \( 1 - 80.1T + 4.48e3T^{2} \)
71 \( 1 + (-42.8 - 42.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (84.0 - 84.0i)T - 5.32e3iT^{2} \)
79 \( 1 + (31.8 + 31.8i)T + 6.24e3iT^{2} \)
83 \( 1 - 46.3T + 6.88e3T^{2} \)
89 \( 1 + 110. iT - 7.92e3T^{2} \)
97 \( 1 + (-57.3 + 57.3i)T - 9.40e3iT^{2} \)
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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

−15.82790135870753035384387659918, −14.19819419251499192845399377276, −13.14840470290187154989781331647, −11.54352146195930020236420395365, −10.33076348007592820491695249857, −9.916326533341962780732699408973, −7.984073724580947551663981698521, −6.87089420697374986377291187530, −5.52171544024330222933718027216, −1.91531148190911712509107658719, 1.01454088253031458862148546103, 4.84892170559117413892649708658, 6.12553512086777460396788743574, 8.212844853794605855301801286781, 9.221626813478002040134133467095, 10.23842348049191540366091426547, 11.09042335487159391827179614689, 12.79342732955840723298356264766, 13.77894571682899637402319889073, 15.88221359265855280062955386339

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