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Properties

Label 2-51-17.16-c1-0-1
Degree $2$
Conductor $51$
Sign $0.242 + 0.970i$
Analytic cond. $0.407237$
Root an. cond. $0.638151$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − i·3-s + 2·4-s − 3i·5-s + 2i·6-s − 2i·7-s − 9-s + 6i·10-s + 5i·11-s − 2i·12-s − 13-s + 4i·14-s − 3·15-s − 4·16-s + (4 − i)17-s + 2·18-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577i·3-s + 4-s − 1.34i·5-s + 0.816i·6-s − 0.755i·7-s − 0.333·9-s + 1.89i·10-s + 1.50i·11-s − 0.577i·12-s − 0.277·13-s + 1.06i·14-s − 0.774·15-s − 16-s + (0.970 − 0.242i)17-s + 0.471·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349043 - 0.272525i\)
\(L(\frac12)\) \(\approx\) \(0.349043 - 0.272525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
17 \( 1 + (-4 + i)T \)
good2 \( 1 + 2T + 2T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−15.84887845474667030444703591069, −14.05191685294401356150237231827, −12.81367126406223898632063433193, −11.83048592172730241922760180118, −10.11702010571057693899380612233, −9.319847335107657312326197133184, −7.981972831200896075642879654744, −7.18842213261785091548349858990, −4.85479155988738852051002926776, −1.30419036797687803621621177953, 3.06339199444238961196745332088, 5.86689401723439675748185924682, 7.48519779331006587519403897841, 8.722653162261079204444829518205, 9.853880726604451897125543567790, 10.79893021295959792211210909271, 11.65908178179317882414980154939, 13.85291752164158867883469039786, 14.90119948851702670542838259770, 16.04530204198886973991244594231

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