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Properties

Label 2-252-252.11-c1-0-27
Degree $2$
Conductor $252$
Sign $0.879 - 0.475i$
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  + (1.41 + 0.107i)2-s + (1.49 + 0.866i)3-s + (1.97 + 0.302i)4-s + (−0.996 − 0.575i)5-s + (2.02 + 1.38i)6-s + (−2.47 + 0.940i)7-s + (2.75 + 0.639i)8-s + (1.49 + 2.59i)9-s + (−1.34 − 0.917i)10-s + (0.0206 + 0.0357i)11-s + (2.70 + 2.16i)12-s + (−2.90 − 5.03i)13-s + (−3.58 + 1.06i)14-s + (−0.995 − 1.72i)15-s + (3.81 + 1.19i)16-s + (−2.56 − 1.48i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0758i)2-s + (0.865 + 0.500i)3-s + (0.988 + 0.151i)4-s + (−0.445 − 0.257i)5-s + (0.825 + 0.564i)6-s + (−0.934 + 0.355i)7-s + (0.974 + 0.225i)8-s + (0.499 + 0.866i)9-s + (−0.424 − 0.290i)10-s + (0.00622 + 0.0107i)11-s + (0.780 + 0.625i)12-s + (−0.806 − 1.39i)13-s + (−0.959 + 0.283i)14-s + (−0.256 − 0.445i)15-s + (0.954 + 0.299i)16-s + (−0.621 − 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.879 - 0.475i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41154 + 0.610650i\)
\(L(\frac12)\) \(\approx\) \(2.41154 + 0.610650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.107i)T \)
3 \( 1 + (-1.49 - 0.866i)T \)
7 \( 1 + (2.47 - 0.940i)T \)
good5 \( 1 + (0.996 + 0.575i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0206 - 0.0357i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.90 + 5.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.56 + 1.48i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.29 + 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.04 - 5.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.45 + 3.72i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.12iT - 31T^{2} \)
37 \( 1 + (-3.94 - 6.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.02 - 1.16i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.07 - 2.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.400T + 47T^{2} \)
53 \( 1 + (-5.08 - 2.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 1.36T + 61T^{2} \)
67 \( 1 - 0.957iT - 67T^{2} \)
71 \( 1 - 1.28T + 71T^{2} \)
73 \( 1 + (-1.36 + 2.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.78iT - 79T^{2} \)
83 \( 1 + (6.87 - 11.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.578 - 0.334i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.04 - 3.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.30632233983783999864021427917, −11.39461763952956887330962391116, −10.11581185302181677399190409856, −9.386743652626142156740580181443, −7.959465702235174844386923772183, −7.28468563785731046837884353995, −5.74848573389661692525115501942, −4.72270494425882756575176414898, −3.46128856747301269452804252366, −2.63860462819158465341131276245, 2.08490977107147419648416571770, 3.43138267674987680183255599299, 4.22217095686337442407382801297, 6.01378985924662694228128393052, 7.07734375149947453513931168675, 7.55704665432475684663528190782, 9.159904450805237030899853803040, 10.04835516180640425897246410905, 11.35320297171560725035344623955, 12.24948773973021597029836928422

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