↑

Properties

Label 2-252-63.31-c2-0-11
Degree $2$
Conductor $252$
Sign $-0.847 + 0.530i$
Analytic cond. $6.86650$
Root an. cond. $2.62040$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.108 + 2.99i)3-s − 2.13i·5-s + (−6.60 − 2.32i)7-s + (−8.97 + 0.652i)9-s − 12.7·11-s + (−4.96 − 2.86i)13-s + (6.40 − 0.232i)15-s + (3.66 + 2.11i)17-s + (−3.77 + 2.17i)19-s + (6.24 − 20.0i)21-s − 27.1·23-s + 20.4·25-s + (−2.93 − 26.8i)27-s + (−14.4 − 25.0i)29-s + (−15.1 + 8.74i)31-s + ⋯
L(s)  = 1  + (0.0362 + 0.999i)3-s − 0.427i·5-s + (−0.943 − 0.331i)7-s + (−0.997 + 0.0725i)9-s − 1.15·11-s + (−0.382 − 0.220i)13-s + (0.427 − 0.0155i)15-s + (0.215 + 0.124i)17-s + (−0.198 + 0.114i)19-s + (0.297 − 0.954i)21-s − 1.17·23-s + 0.817·25-s + (−0.108 − 0.994i)27-s + (−0.499 − 0.865i)29-s + (−0.488 + 0.282i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.847 + 0.530i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ -0.847 + 0.530i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00172943 - 0.00602456i\)
\(L(\frac12)\) \(\approx\) \(0.00172943 - 0.00602456i\)
\(L(2)\) 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 \( 1 + (-0.108 - 2.99i)T \)
7 \( 1 + (6.60 + 2.32i)T \)
good5 \( 1 + 2.13iT - 25T^{2} \)
11 \( 1 + 12.7T + 121T^{2} \)
13 \( 1 + (4.96 + 2.86i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-3.66 - 2.11i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.77 - 2.17i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + 27.1T + 529T^{2} \)
29 \( 1 + (14.4 + 25.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (15.1 - 8.74i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (2.21 + 3.84i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-2.24 - 1.29i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-26.5 - 46.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (68.7 + 39.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (18.7 - 32.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.5 - 12.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.6 - 13.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-64.7 - 112. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 68.2T + 5.04e3T^{2} \)
73 \( 1 + (88.2 + 50.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (2.17 - 3.77i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (38.3 - 22.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (84.0 - 48.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-139. + 80.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−11.24004615084839408730540758833, −10.21907300180281203875505115445, −9.769429568164434127837087706771, −8.627490602908347920377300010556, −7.63069769077182496019077703158, −6.11452620360431713539664455860, −5.12110092602623996508467741608, −3.96361107551784061667333731426, −2.74324986304408755351174514267, −0.00296749095068759067368500309, 2.22970206136903376381830946009, 3.28837594097474759621208762279, 5.28747000728067554047747544376, 6.33116449316828585012447560587, 7.20756604702899867712607945917, 8.138798084385775158947781906427, 9.283636868463086746652548526760, 10.35518830309075480525731122113, 11.35324288893702696980796005876, 12.50309941479922269794517342284

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