↑

Properties

Label 2-252-63.58-c1-0-6
Degree $2$
Conductor $252$
Sign $-0.109 + 0.993i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (0.799 − 1.53i)3-s + (−0.483 − 0.838i)5-s + (−1.52 − 2.16i)7-s + (−1.72 − 2.45i)9-s + (−0.364 + 0.630i)11-s + (1.81 − 3.13i)13-s + (−1.67 + 0.0731i)15-s + (3.49 + 6.06i)17-s + (−0.348 + 0.602i)19-s + (−4.54 + 0.604i)21-s + (−3.21 − 5.57i)23-s + (2.03 − 3.51i)25-s + (−5.15 + 0.678i)27-s + (3.34 + 5.79i)29-s + 9.16·31-s + ⋯
L(s)  = 1  + (0.461 − 0.887i)3-s + (−0.216 − 0.374i)5-s + (−0.574 − 0.818i)7-s + (−0.573 − 0.819i)9-s + (−0.109 + 0.190i)11-s + (0.502 − 0.869i)13-s + (−0.432 + 0.0188i)15-s + (0.848 + 1.47i)17-s + (−0.0798 + 0.138i)19-s + (−0.991 + 0.131i)21-s + (−0.671 − 1.16i)23-s + (0.406 − 0.703i)25-s + (−0.991 + 0.130i)27-s + (0.621 + 1.07i)29-s + 1.64·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.109 + 0.993i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848672 - 0.947326i\)
\(L(\frac12)\) \(\approx\) \(0.848672 - 0.947326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.799 + 1.53i)T \)
7 \( 1 + (1.52 + 2.16i)T \)
good5 \( 1 + (0.483 + 0.838i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.364 - 0.630i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.81 + 3.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.49 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.348 - 0.602i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.21 + 5.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.34 - 5.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + (-0.854 + 1.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.62 - 6.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.348 + 0.602i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.66T + 47T^{2} \)
53 \( 1 + (-2.05 - 3.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.77T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 5.82T + 67T^{2} \)
71 \( 1 - 0.304T + 71T^{2} \)
73 \( 1 + (-5.33 - 9.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.23T + 79T^{2} \)
83 \( 1 + (-0.618 - 1.07i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.78 - 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.32 - 2.30i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.31078978325729117761270673920, −10.68606426224548428817833018743, −9.992773668322115107481953065902, −8.457735909331771041345505460127, −8.069912045147653283452640231223, −6.80455862168897367246620749637, −5.95525965324742965509080303526, −4.19011501865583434923502474607, −2.97097173802909780009815355855, −1.04619314307125930409596184284, 2.62592708152449770135714455476, 3.63306224985434148952944158498, 5.00957326068326539872345386249, 6.13905820930726546403986184493, 7.48607487243674884661376155710, 8.641541313302524160384519850191, 9.466230194885863948413579833380, 10.17076663407171781271395557278, 11.46998460568059585376589050578, 11.94929881913595447687100581724

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