↑

Properties

Label 2-51-17.9-c1-0-0
Degree $2$
Conductor $51$
Sign $0.0340 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 1.30i)2-s + (0.923 + 0.382i)3-s − 1.41i·4-s + (−0.617 + 1.49i)5-s + (−1.70 + 0.707i)6-s + (0.0582 + 0.140i)7-s + (−0.765 − 0.765i)8-s + (0.707 + 0.707i)9-s + (−1.14 − 2.75i)10-s + (4.64 − 1.92i)11-s + (0.541 − 1.30i)12-s − 3.94i·13-s + (−0.259 − 0.107i)14-s + (−1.14 + 1.14i)15-s + 4.82·16-s + (−1.26 − 3.92i)17-s + ⋯
L(s)  = 1  + (−0.923 + 0.923i)2-s + (0.533 + 0.220i)3-s − 0.707i·4-s + (−0.276 + 0.666i)5-s + (−0.696 + 0.288i)6-s + (0.0220 + 0.0531i)7-s + (−0.270 − 0.270i)8-s + (0.235 + 0.235i)9-s + (−0.360 − 0.870i)10-s + (1.40 − 0.580i)11-s + (0.156 − 0.377i)12-s − 1.09i·13-s + (−0.0694 − 0.0287i)14-s + (−0.294 + 0.294i)15-s + 1.20·16-s + (−0.307 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.0340 - 0.999i$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ 0.0340 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447548 + 0.432569i\)
\(L(\frac12)\) \(\approx\) \(0.447548 + 0.432569i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (1.26 + 3.92i)T \)
good2 \( 1 + (1.30 - 1.30i)T - 2iT^{2} \)
5 \( 1 + (0.617 - 1.49i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.0582 - 0.140i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-4.64 + 1.92i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.94iT - 13T^{2} \)
19 \( 1 + (4.65 - 4.65i)T - 19iT^{2} \)
23 \( 1 + (3.18 - 1.31i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.858 + 2.07i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.37 + 0.985i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (9.86 + 4.08i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.105 + 0.255i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4.48 - 4.48i)T + 43iT^{2} \)
47 \( 1 - 9.82iT - 47T^{2} \)
53 \( 1 + (1.50 - 1.50i)T - 53iT^{2} \)
59 \( 1 + (0.936 + 0.936i)T + 59iT^{2} \)
61 \( 1 + (3.16 + 7.64i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.10T + 67T^{2} \)
71 \( 1 + (-6.04 - 2.50i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.18 + 7.69i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.491 + 0.203i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.55 - 1.55i)T - 83iT^{2} \)
89 \( 1 - 7.64iT - 89T^{2} \)
97 \( 1 + (1.38 - 3.33i)T + (-68.5 - 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.83738853543287608459046935588, −14.93699233770025408665787808971, −14.08582176565776591145797471349, −12.36059351176860197789471765519, −10.80638519653093845372161762244, −9.512517295538756329835995129002, −8.478117802498085807117723435461, −7.38998183271144685486889467116, −6.14261388020580226434661658745, −3.54805267326852940510808180226, 1.82499421747110239944157808539, 4.17356925089384445664280537309, 6.74529802073424350925271447681, 8.650550888697843536537430873480, 9.034067292067672284149515472640, 10.44784168227936021535196362079, 11.79281254141008659237028204688, 12.55037405901413777101949636371, 14.11854574665543146573637760157, 15.18083044552725459758448213370

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