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Properties

Label 2-51-51.47-c2-0-1
Degree $2$
Conductor $51$
Sign $-0.540 - 0.841i$
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  − 3.37·2-s + (−1.14 − 2.77i)3-s + 7.40·4-s + (−1.81 + 1.81i)5-s + (3.87 + 9.35i)6-s + (−8.06 + 8.06i)7-s − 11.4·8-s + (−6.36 + 6.36i)9-s + (6.14 − 6.14i)10-s + (5.55 + 5.55i)11-s + (−8.49 − 20.5i)12-s − 3.00·13-s + (27.2 − 27.2i)14-s + (7.13 + 2.95i)15-s + 9.16·16-s + (−11.3 + 12.6i)17-s + ⋯
L(s)  = 1  − 1.68·2-s + (−0.382 − 0.923i)3-s + 1.85·4-s + (−0.363 + 0.363i)5-s + (0.645 + 1.55i)6-s + (−1.15 + 1.15i)7-s − 1.43·8-s + (−0.707 + 0.707i)9-s + (0.614 − 0.614i)10-s + (0.505 + 0.505i)11-s + (−0.707 − 1.70i)12-s − 0.230·13-s + (1.94 − 1.94i)14-s + (0.475 + 0.197i)15-s + 0.572·16-s + (−0.666 + 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.540 - 0.841i$
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: \((2,\ 51,\ (\ :1),\ -0.540 - 0.841i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0700426 + 0.128274i\)
\(L(\frac12)\) \(\approx\) \(0.0700426 + 0.128274i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.14 + 2.77i)T \)
17 \( 1 + (11.3 - 12.6i)T \)
good2 \( 1 + 3.37T + 4T^{2} \)
5 \( 1 + (1.81 - 1.81i)T - 25iT^{2} \)
7 \( 1 + (8.06 - 8.06i)T - 49iT^{2} \)
11 \( 1 + (-5.55 - 5.55i)T + 121iT^{2} \)
13 \( 1 + 3.00T + 169T^{2} \)
19 \( 1 + 18.5iT - 361T^{2} \)
23 \( 1 + (29.6 + 29.6i)T + 529iT^{2} \)
29 \( 1 + (19.6 - 19.6i)T - 841iT^{2} \)
31 \( 1 + (9.81 + 9.81i)T + 961iT^{2} \)
37 \( 1 + (-1.50 - 1.50i)T + 1.36e3iT^{2} \)
41 \( 1 + (-26.4 - 26.4i)T + 1.68e3iT^{2} \)
43 \( 1 - 10.3iT - 1.84e3T^{2} \)
47 \( 1 + 28.9iT - 2.20e3T^{2} \)
53 \( 1 + 7.41T + 2.80e3T^{2} \)
59 \( 1 - 22.7T + 3.48e3T^{2} \)
61 \( 1 + (49.7 - 49.7i)T - 3.72e3iT^{2} \)
67 \( 1 - 18.0T + 4.48e3T^{2} \)
71 \( 1 + (12.6 - 12.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (-84.8 - 84.8i)T + 5.32e3iT^{2} \)
79 \( 1 + (19.0 - 19.0i)T - 6.24e3iT^{2} \)
83 \( 1 - 92.0T + 6.88e3T^{2} \)
89 \( 1 + 12.6iT - 7.92e3T^{2} \)
97 \( 1 + (8.53 + 8.53i)T + 9.40e3iT^{2} \)
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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

−16.04840903285524077292912008212, −14.95703977161720585450095708483, −12.94119716264589962472167282283, −11.95528423085432051858800119866, −10.88929999833664045266243655355, −9.490765276098300514115344651265, −8.485940449878477821374059799519, −7.10961505322389655079618811402, −6.24648169092433496782794047842, −2.32917241639422969793909616785, 0.24677731281228048928395084702, 3.84798756616340850923630319535, 6.31955479168377785353800184322, 7.73795810861957660671381186561, 9.218830577266152829420127887177, 9.900249044008014351794629383888, 10.88742771499567745127588723765, 12.00546686652650611742497845501, 13.89084854769617322605569956026, 15.72893343991412351501795923689

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