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Properties

Label 2-51-17.4-c1-0-3
Degree $2$
Conductor $51$
Sign $-0.447 + 0.894i$
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  − 2.63i·2-s + (−0.707 + 0.707i)3-s − 4.95·4-s + (1.57 − 1.57i)5-s + (1.86 + 1.86i)6-s + (1.63 + 1.63i)7-s + 7.77i·8-s − 1.00i·9-s + (−4.14 − 4.14i)10-s + (1.37 + 1.37i)11-s + (3.50 − 3.50i)12-s − 1.32·13-s + (4.31 − 4.31i)14-s + 2.22i·15-s + 10.6·16-s + (−3.88 − 1.37i)17-s + ⋯
L(s)  = 1  − 1.86i·2-s + (−0.408 + 0.408i)3-s − 2.47·4-s + (0.702 − 0.702i)5-s + (0.761 + 0.761i)6-s + (0.618 + 0.618i)7-s + 2.75i·8-s − 0.333i·9-s + (−1.31 − 1.31i)10-s + (0.415 + 0.415i)11-s + (1.01 − 1.01i)12-s − 0.366·13-s + (1.15 − 1.15i)14-s + 0.573i·15-s + 2.65·16-s + (−0.942 − 0.334i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391830 - 0.633959i\)
\(L(\frac12)\) \(\approx\) \(0.391830 - 0.633959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (3.88 + 1.37i)T \)
good2 \( 1 + 2.63iT - 2T^{2} \)
5 \( 1 + (-1.57 + 1.57i)T - 5iT^{2} \)
7 \( 1 + (-1.63 - 1.63i)T + 7iT^{2} \)
11 \( 1 + (-1.37 - 1.37i)T + 11iT^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
19 \( 1 - 5.95iT - 19T^{2} \)
23 \( 1 + (3.37 + 3.37i)T + 23iT^{2} \)
29 \( 1 + (0.122 - 0.122i)T - 29iT^{2} \)
31 \( 1 + (-0.314 + 0.314i)T - 31iT^{2} \)
37 \( 1 + (-3.75 + 3.75i)T - 37iT^{2} \)
41 \( 1 + (2.84 + 2.84i)T + 41iT^{2} \)
43 \( 1 + 8.33iT - 43T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 + 0.727iT - 53T^{2} \)
59 \( 1 - 11.7iT - 59T^{2} \)
61 \( 1 + (-6.14 - 6.14i)T + 61iT^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + (-4.31 + 4.31i)T - 71iT^{2} \)
73 \( 1 + (-7.46 + 7.46i)T - 73iT^{2} \)
79 \( 1 + (-0.493 - 0.493i)T + 79iT^{2} \)
83 \( 1 - 8.52iT - 83T^{2} \)
89 \( 1 + 5.40T + 89T^{2} \)
97 \( 1 + (-3.33 + 3.33i)T - 97iT^{2} \)
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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

−14.80507586071677193458404172844, −13.62950785091286031428341907347, −12.41716701890655270334470928698, −11.80901226468877396334934198709, −10.54235309855242013906812918324, −9.554823557058639491988905506893, −8.643315163566147680173303399821, −5.46466237600052477189876580057, −4.27051790441278271932000630544, −1.98266399988151318784730276970, 4.68837996991755866303255918275, 6.17155260356155897279096232802, 6.95911035252782178753972823013, 8.168609419303141741300458952264, 9.633036731700231880081823163898, 11.16993505034409403267290017924, 13.20745332331958750012771394407, 13.92709930258729998113744150401, 14.77440730662223007601072534291, 15.92186119347685637198163732574

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