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Properties

Label 2-252-252.115-c1-0-1
Degree $2$
Conductor $252$
Sign $-0.155 - 0.987i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0863 − 1.41i)2-s + (−0.595 + 1.62i)3-s + (−1.98 − 0.243i)4-s + (−1.73 − 1.00i)5-s + (2.24 + 0.980i)6-s + (1.27 + 2.31i)7-s + (−0.515 + 2.78i)8-s + (−2.29 − 1.93i)9-s + (−1.56 + 2.36i)10-s + (−4.35 + 2.51i)11-s + (1.57 − 3.08i)12-s + (−3.29 + 1.90i)13-s + (3.38 − 1.60i)14-s + (2.66 − 2.22i)15-s + (3.88 + 0.968i)16-s + (1.61 + 0.932i)17-s + ⋯
L(s)  = 1  + (0.0610 − 0.998i)2-s + (−0.343 + 0.939i)3-s + (−0.992 − 0.121i)4-s + (−0.775 − 0.447i)5-s + (0.916 + 0.400i)6-s + (0.482 + 0.875i)7-s + (−0.182 + 0.983i)8-s + (−0.763 − 0.645i)9-s + (−0.494 + 0.746i)10-s + (−1.31 + 0.758i)11-s + (0.455 − 0.890i)12-s + (−0.913 + 0.527i)13-s + (0.903 − 0.428i)14-s + (0.686 − 0.574i)15-s + (0.970 + 0.242i)16-s + (0.391 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290092 + 0.339202i\)
\(L(\frac12)\) \(\approx\) \(0.290092 + 0.339202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0863 + 1.41i)T \)
3 \( 1 + (0.595 - 1.62i)T \)
7 \( 1 + (-1.27 - 2.31i)T \)
good5 \( 1 + (1.73 + 1.00i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.35 - 2.51i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.29 - 1.90i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.61 - 0.932i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.12 - 5.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.43 + 3.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.23 - 2.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.11T + 31T^{2} \)
37 \( 1 + (-0.559 - 0.968i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.39 + 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.79 + 3.92i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 + (-2.80 + 4.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 2.11iT - 61T^{2} \)
67 \( 1 - 12.0iT - 67T^{2} \)
71 \( 1 - 3.84iT - 71T^{2} \)
73 \( 1 + (0.499 + 0.288i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.95iT - 79T^{2} \)
83 \( 1 + (5.34 - 9.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.13 - 2.96i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.0 - 8.10i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.06982294358005972865778396866, −11.57464222723372996412085021308, −10.27763965296611572061034497982, −9.830291253981144441850377583718, −8.594836505983680781642602762835, −7.85776473954119572116335973345, −5.58694221136292363988032093372, −4.87514891840294140556578976168, −3.90502708759130919607585866231, −2.37178937650945366341581530041, 0.35229863503835513501926961635, 3.13314125909824741606649300910, 4.80759147393975493830927906931, 5.74303178272229587574558598790, 7.11608839632643824713335900287, 7.68596422512021384463318467473, 8.144129905793315523881952701518, 9.852232666668097789449541729478, 10.98854339877110044121683829989, 11.80867066180031560161501094021

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