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Properties

Label 2-252-63.38-c1-0-1
Degree $2$
Conductor $252$
Sign $-0.753 - 0.657i$
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  + (−0.134 + 1.72i)3-s + (−0.842 + 1.45i)5-s + (−2.27 + 1.34i)7-s + (−2.96 − 0.464i)9-s + (3.38 − 1.95i)11-s + (−5.24 + 3.02i)13-s + (−2.40 − 1.65i)15-s + (−0.201 + 0.348i)17-s + (−0.145 + 0.0840i)19-s + (−2.01 − 4.11i)21-s + (7.69 + 4.44i)23-s + (1.07 + 1.86i)25-s + (1.20 − 5.05i)27-s + (−6.15 − 3.55i)29-s + 6.28i·31-s + ⋯
L(s)  = 1  + (−0.0776 + 0.996i)3-s + (−0.376 + 0.652i)5-s + (−0.861 + 0.507i)7-s + (−0.987 − 0.154i)9-s + (1.01 − 0.588i)11-s + (−1.45 + 0.839i)13-s + (−0.621 − 0.426i)15-s + (−0.0488 + 0.0845i)17-s + (−0.0334 + 0.0192i)19-s + (−0.439 − 0.898i)21-s + (1.60 + 0.926i)23-s + (0.215 + 0.373i)25-s + (0.230 − 0.972i)27-s + (−1.14 − 0.659i)29-s + 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.753 - 0.657i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.298106 + 0.794588i\)
\(L(\frac12)\) \(\approx\) \(0.298106 + 0.794588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.134 - 1.72i)T \)
7 \( 1 + (2.27 - 1.34i)T \)
good5 \( 1 + (0.842 - 1.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.38 + 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.24 - 3.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.201 - 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.145 - 0.0840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.69 - 4.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.15 + 3.55i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.28iT - 31T^{2} \)
37 \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.64 - 2.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.80 + 3.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.76T + 47T^{2} \)
53 \( 1 + (-4.94 - 2.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.50T + 59T^{2} \)
61 \( 1 + 5.12iT - 61T^{2} \)
67 \( 1 + 5.91T + 67T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (6.05 + 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.20T + 79T^{2} \)
83 \( 1 + (-0.181 + 0.314i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.38 + 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.508 + 0.293i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.04218478356540436274484190691, −11.48361248315477070992668843032, −10.50558265355106434961036931246, −9.333588188883198176428912315788, −9.074739686246493328653013646809, −7.33388483061163121789373006482, −6.38081125210787425363435547436, −5.15717732078547152628017613079, −3.81789562999822658413922938498, −2.86005390938659416997892367579, 0.67636431417076130490966216984, 2.65161382940832853329034962397, 4.27923791973941173219471540970, 5.61912026342537212760678625809, 6.97278387853150503933548309396, 7.41839826010296016936151653708, 8.754860277729708068138189618913, 9.608459446813869630057002249973, 10.85071224778795225390480321002, 12.01796667084658248208453586966

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