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Properties

Label 2-252-252.139-c1-0-1
Degree $2$
Conductor $252$
Sign $0.386 - 0.922i$
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  + (−1.33 + 0.471i)2-s + (−1.61 − 0.633i)3-s + (1.55 − 1.25i)4-s + (−3.16 − 1.82i)5-s + (2.44 + 0.0856i)6-s + (−2.26 + 1.37i)7-s + (−1.48 + 2.40i)8-s + (2.19 + 2.04i)9-s + (5.08 + 0.947i)10-s + (1.96 − 1.13i)11-s + (−3.30 + 1.03i)12-s + (3.84 + 2.21i)13-s + (2.36 − 2.89i)14-s + (3.94 + 4.95i)15-s + (0.844 − 3.90i)16-s + 3.25i·17-s + ⋯
L(s)  = 1  + (−0.942 + 0.333i)2-s + (−0.930 − 0.365i)3-s + (0.778 − 0.628i)4-s + (−1.41 − 0.818i)5-s + (0.999 + 0.0349i)6-s + (−0.854 + 0.518i)7-s + (−0.524 + 0.851i)8-s + (0.732 + 0.680i)9-s + (1.60 + 0.299i)10-s + (0.592 − 0.342i)11-s + (−0.953 + 0.299i)12-s + (1.06 + 0.615i)13-s + (0.633 − 0.773i)14-s + (1.01 + 1.28i)15-s + (0.211 − 0.977i)16-s + 0.789i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282347 + 0.187735i\)
\(L(\frac12)\) \(\approx\) \(0.282347 + 0.187735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.471i)T \)
3 \( 1 + (1.61 + 0.633i)T \)
7 \( 1 + (2.26 - 1.37i)T \)
good5 \( 1 + (3.16 + 1.82i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.96 + 1.13i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.84 - 2.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.25iT - 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + (1.52 + 0.879i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.73 - 6.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.37 - 5.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.82T + 37T^{2} \)
41 \( 1 + (2.24 + 1.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.27 + 3.62i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.95 - 6.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + (-0.278 + 0.482i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.335 - 0.193i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.07 + 1.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.43iT - 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 + (-2.20 + 1.27i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.18 - 7.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.43iT - 89T^{2} \)
97 \( 1 + (-5.47 + 3.15i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.10983596809449005683559345494, −11.29594625576649741231201439516, −10.45008506757787422647019746783, −8.978692964593597569179590857989, −8.492297171183165676306852864088, −7.23452057388697372162905514094, −6.41936811641972487596372448943, −5.33987031879823151320957026778, −3.75620686643733337211929209936, −1.20015484379950667048516131765, 0.48810473504796708504890481959, 3.32121296031560148237488727328, 4.04524710535352986343903408348, 6.16959690406620098850710519370, 7.03949278521053081558751181193, 7.77235568993896553770917917627, 9.207188786819945153885668483802, 10.20096992747564431616390340206, 10.85571280979442089113751116458, 11.70235884044943560417744378002

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