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Properties

Label 2-51-51.26-c2-0-3
Degree $2$
Conductor $51$
Sign $0.447 - 0.894i$
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  + (1.70 + 1.70i)2-s + (2.12 + 2.12i)3-s + 1.82i·4-s + (−4.82 − 2i)5-s + 7.24i·6-s + (−3.82 − 9.24i)7-s + (3.70 − 3.70i)8-s + 8.99i·9-s + (−4.82 − 11.6i)10-s + (3.29 + 7.94i)11-s + (−3.87 + 3.87i)12-s + 14.8i·13-s + (9.24 − 22.3i)14-s + (−5.99 − 14.4i)15-s + 19.9·16-s + (−4.60 − 16.3i)17-s + ⋯
L(s)  = 1  + (0.853 + 0.853i)2-s + (0.707 + 0.707i)3-s + 0.457i·4-s + (−0.965 − 0.400i)5-s + 1.20i·6-s + (−0.546 − 1.32i)7-s + (0.463 − 0.463i)8-s + 0.999i·9-s + (−0.482 − 1.16i)10-s + (0.299 + 0.722i)11-s + (−0.323 + 0.323i)12-s + 1.14i·13-s + (0.660 − 1.59i)14-s + (−0.399 − 0.965i)15-s + 1.24·16-s + (−0.270 − 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49005 + 0.920066i\)
\(L(\frac12)\) \(\approx\) \(1.49005 + 0.920066i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.12 - 2.12i)T \)
17 \( 1 + (4.60 + 16.3i)T \)
good2 \( 1 + (-1.70 - 1.70i)T + 4iT^{2} \)
5 \( 1 + (4.82 + 2i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (3.82 + 9.24i)T + (-34.6 + 34.6i)T^{2} \)
11 \( 1 + (-3.29 - 7.94i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 - 14.8iT - 169T^{2} \)
19 \( 1 + (12.8 - 12.8i)T - 361iT^{2} \)
23 \( 1 + (5.72 + 13.8i)T + (-374. + 374. i)T^{2} \)
29 \( 1 + (-18.3 - 7.61i)T + (594. + 594. i)T^{2} \)
31 \( 1 + (9.72 + 4.02i)T + (679. + 679. i)T^{2} \)
37 \( 1 + (-8.82 - 3.65i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (-3.77 + 1.56i)T + (1.18e3 - 1.18e3i)T^{2} \)
43 \( 1 + (-22.5 - 22.5i)T + 1.84e3iT^{2} \)
47 \( 1 - 87.5T + 2.20e3T^{2} \)
53 \( 1 + (17.3 + 17.3i)T + 2.80e3iT^{2} \)
59 \( 1 + (70.7 - 70.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (44.1 + 106. i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 - 48.1T + 4.48e3T^{2} \)
71 \( 1 + (16.4 - 39.8i)T + (-3.56e3 - 3.56e3i)T^{2} \)
73 \( 1 + (-4.16 + 10.0i)T + (-3.76e3 - 3.76e3i)T^{2} \)
79 \( 1 + (-54.8 + 22.7i)T + (4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (-86.7 - 86.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 0.159T + 7.92e3T^{2} \)
97 \( 1 + (20.4 - 49.2i)T + (-6.65e3 - 6.65e3i)T^{2} \)
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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

−15.45244295735913469110902636265, −14.33217324936028210437595003920, −13.71822755337192363318460745779, −12.39237772613980672224576118221, −10.66005903984035983080605135259, −9.447085662121351815448036624255, −7.79374148899496923939897373627, −6.78745823455491607112742615716, −4.50265992540263428593831574780, −4.01512286321599360167434665994, 2.64063916672503786242798039205, 3.69310761888022620806299318843, 5.98226137279365479393565455070, 7.80161117378704168819170930107, 8.847220273071288707400714683596, 10.85224987712844918871598306841, 12.01848279135448282708526596821, 12.63358636706005115232610392882, 13.62983389787773520164796083874, 14.96760402256604545371530598178

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