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Properties

Label 2-51-17.4-c5-0-9
Degree $2$
Conductor $51$
Sign $-0.482 + 0.875i$
Analytic cond. $8.17957$
Root an. cond. $2.85999$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.59i·2-s + (−6.36 + 6.36i)3-s − 11.4·4-s + (50.2 − 50.2i)5-s + (41.9 + 41.9i)6-s + (116. + 116. i)7-s − 135. i·8-s − 81i·9-s + (−330. − 330. i)10-s + (−467. − 467. i)11-s + (72.7 − 72.7i)12-s + 691.·13-s + (768. − 768. i)14-s + 638. i·15-s − 1.25e3·16-s + (−1.10e3 − 443. i)17-s + ⋯
L(s)  = 1  − 1.16i·2-s + (−0.408 + 0.408i)3-s − 0.357·4-s + (0.898 − 0.898i)5-s + (0.475 + 0.475i)6-s + (0.898 + 0.898i)7-s − 0.748i·8-s − 0.333i·9-s + (−1.04 − 1.04i)10-s + (−1.16 − 1.16i)11-s + (0.145 − 0.145i)12-s + 1.13·13-s + (1.04 − 1.04i)14-s + 0.733i·15-s − 1.22·16-s + (−0.928 − 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.482 + 0.875i$
Analytic conductor: \(8.17957\)
Root analytic conductor: \(2.85999\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :5/2),\ -0.482 + 0.875i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.924278 - 1.56535i\)
\(L(\frac12)\) \(\approx\) \(0.924278 - 1.56535i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.36 - 6.36i)T \)
17 \( 1 + (1.10e3 + 443. i)T \)
good2 \( 1 + 6.59iT - 32T^{2} \)
5 \( 1 + (-50.2 + 50.2i)T - 3.12e3iT^{2} \)
7 \( 1 + (-116. - 116. i)T + 1.68e4iT^{2} \)
11 \( 1 + (467. + 467. i)T + 1.61e5iT^{2} \)
13 \( 1 - 691.T + 3.71e5T^{2} \)
19 \( 1 + 1.48e3iT - 2.47e6T^{2} \)
23 \( 1 + (-2.07e3 - 2.07e3i)T + 6.43e6iT^{2} \)
29 \( 1 + (-5.35e3 + 5.35e3i)T - 2.05e7iT^{2} \)
31 \( 1 + (2.86e3 - 2.86e3i)T - 2.86e7iT^{2} \)
37 \( 1 + (-5.42e3 + 5.42e3i)T - 6.93e7iT^{2} \)
41 \( 1 + (-1.02e4 - 1.02e4i)T + 1.15e8iT^{2} \)
43 \( 1 - 1.29e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.08e4T + 2.29e8T^{2} \)
53 \( 1 - 1.98e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.31e4iT - 7.14e8T^{2} \)
61 \( 1 + (6.24e3 + 6.24e3i)T + 8.44e8iT^{2} \)
67 \( 1 - 5.78e4T + 1.35e9T^{2} \)
71 \( 1 + (2.70e4 - 2.70e4i)T - 1.80e9iT^{2} \)
73 \( 1 + (644. - 644. i)T - 2.07e9iT^{2} \)
79 \( 1 + (-2.03e4 - 2.03e4i)T + 3.07e9iT^{2} \)
83 \( 1 - 3.53e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.00e4T + 5.58e9T^{2} \)
97 \( 1 + (2.74e4 - 2.74e4i)T - 8.58e9iT^{2} \)
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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

−13.55050110877161596137243252699, −12.92791796490909074749974019280, −11.38438896703138864156143123310, −11.00597659159181701726476389827, −9.465830599599670535352770666460, −8.543358022376437282270475147654, −5.96208853321877932217188197272, −4.81262520788915875102227099963, −2.66694506396851441001901669744, −1.04825911590987139362199847515, 2.01442289871166526894113681564, 4.92135001189599011732965531453, 6.30340007569462707437237098741, 7.16347462640960167384436533297, 8.268293057857774899055411845242, 10.42442102105068209989621127633, 11.03438405788699512808875628716, 12.97551409534860415440033895525, 14.04607184690471584424053370225, 14.80273898937052546648896149027

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