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Properties

Label 2-252-12.11-c3-0-15
Degree $2$
Conductor $252$
Sign $0.253 - 0.967i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.968 + 2.65i)2-s + (−6.12 + 5.14i)4-s − 12.6i·5-s + 7i·7-s + (−19.6 − 11.2i)8-s + (33.6 − 12.2i)10-s + 54.6·11-s + 20.4·13-s + (−18.6 + 6.77i)14-s + (11.0 − 63.0i)16-s + 134. i·17-s + 9.73i·19-s + (65.1 + 77.4i)20-s + (52.9 + 145. i)22-s + 161.·23-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.765 + 0.643i)4-s − 1.13i·5-s + 0.377i·7-s + (−0.866 − 0.498i)8-s + (1.06 − 0.387i)10-s + 1.49·11-s + 0.435·13-s + (−0.355 + 0.129i)14-s + (0.171 − 0.985i)16-s + 1.92i·17-s + 0.117i·19-s + (0.728 + 0.866i)20-s + (0.513 + 1.40i)22-s + 1.46·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.253 - 0.967i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.103471759\)
\(L(\frac12)\) \(\approx\) \(2.103471759\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.968 - 2.65i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 + 12.6iT - 125T^{2} \)
11 \( 1 - 54.6T + 1.33e3T^{2} \)
13 \( 1 - 20.4T + 2.19e3T^{2} \)
17 \( 1 - 134. iT - 4.91e3T^{2} \)
19 \( 1 - 9.73iT - 6.85e3T^{2} \)
23 \( 1 - 161.T + 1.21e4T^{2} \)
29 \( 1 - 216. iT - 2.43e4T^{2} \)
31 \( 1 + 240. iT - 2.97e4T^{2} \)
37 \( 1 - 72.2T + 5.06e4T^{2} \)
41 \( 1 + 284. iT - 6.89e4T^{2} \)
43 \( 1 - 402. iT - 7.95e4T^{2} \)
47 \( 1 - 286.T + 1.03e5T^{2} \)
53 \( 1 - 236. iT - 1.48e5T^{2} \)
59 \( 1 + 445.T + 2.05e5T^{2} \)
61 \( 1 + 230.T + 2.26e5T^{2} \)
67 \( 1 + 697. iT - 3.00e5T^{2} \)
71 \( 1 - 786.T + 3.57e5T^{2} \)
73 \( 1 + 400.T + 3.89e5T^{2} \)
79 \( 1 + 119. iT - 4.93e5T^{2} \)
83 \( 1 - 1.30e3T + 5.71e5T^{2} \)
89 \( 1 - 557. iT - 7.04e5T^{2} \)
97 \( 1 - 149.T + 9.12e5T^{2} \)
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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

−12.28839948002654029313464936563, −10.96406548059888283755995767926, −9.246381532043135222349010015445, −8.913264270513893359590618557457, −7.958237916403170280107482609597, −6.58894078820172479297972508581, −5.77562052682808028914254629948, −4.60233322187694249108875478443, −3.65087609304722677319649123792, −1.23227077816894202597298529877, 0.971149896330171470788489566358, 2.69836070897661500238188503671, 3.65587381375519703259023137860, 4.89339312517970968007868305997, 6.38881960888264959091990453745, 7.18290000153863825255949204965, 8.890115932286247060616615423148, 9.643262466622897840125175033699, 10.69237818956246866387967214841, 11.40004638680635275909141753094

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