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Properties

Label 2-252-12.11-c3-0-12
Degree $2$
Conductor $252$
Sign $-0.851 - 0.523i$
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  + (2.36 + 1.55i)2-s + (3.14 + 7.35i)4-s + 8.56i·5-s + 7i·7-s + (−4.02 + 22.2i)8-s + (−13.3 + 20.2i)10-s − 21.7·11-s − 3.31·13-s + (−10.9 + 16.5i)14-s + (−44.1 + 46.2i)16-s + 12.8i·17-s − 7.76i·19-s + (−62.9 + 26.9i)20-s + (−51.2 − 33.8i)22-s − 74.9·23-s + ⋯
L(s)  = 1  + (0.834 + 0.550i)2-s + (0.393 + 0.919i)4-s + 0.765i·5-s + 0.377i·7-s + (−0.177 + 0.984i)8-s + (−0.421 + 0.639i)10-s − 0.594·11-s − 0.0707·13-s + (−0.208 + 0.315i)14-s + (−0.690 + 0.723i)16-s + 0.182i·17-s − 0.0937i·19-s + (−0.704 + 0.301i)20-s + (−0.496 − 0.327i)22-s − 0.679·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.851 - 0.523i$
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: \((2,\ 252,\ (\ :3/2),\ -0.851 - 0.523i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.384178182\)
\(L(\frac12)\) \(\approx\) \(2.384178182\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.36 - 1.55i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 - 8.56iT - 125T^{2} \)
11 \( 1 + 21.7T + 1.33e3T^{2} \)
13 \( 1 + 3.31T + 2.19e3T^{2} \)
17 \( 1 - 12.8iT - 4.91e3T^{2} \)
19 \( 1 + 7.76iT - 6.85e3T^{2} \)
23 \( 1 + 74.9T + 1.21e4T^{2} \)
29 \( 1 - 157. iT - 2.43e4T^{2} \)
31 \( 1 + 15.8iT - 2.97e4T^{2} \)
37 \( 1 + 14.2T + 5.06e4T^{2} \)
41 \( 1 - 474. iT - 6.89e4T^{2} \)
43 \( 1 + 375. iT - 7.95e4T^{2} \)
47 \( 1 - 256.T + 1.03e5T^{2} \)
53 \( 1 - 474. iT - 1.48e5T^{2} \)
59 \( 1 - 686.T + 2.05e5T^{2} \)
61 \( 1 - 117.T + 2.26e5T^{2} \)
67 \( 1 + 421. iT - 3.00e5T^{2} \)
71 \( 1 + 5.74T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 + 683. iT - 4.93e5T^{2} \)
83 \( 1 + 130.T + 5.71e5T^{2} \)
89 \( 1 + 205. iT - 7.04e5T^{2} \)
97 \( 1 - 1.71e3T + 9.12e5T^{2} \)
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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

−12.15986746941048446456280073723, −11.18587007301027599830315608914, −10.30370118191256473925475831868, −8.856353156341123708429934163766, −7.81559674944286753740776669494, −6.88431809452054482255773249511, −5.92595629747953187787972192844, −4.86091454089062356510911213171, −3.46967819286388210636937526824, −2.38345843117996080512990209978, 0.69841534751079519401162105482, 2.27489960309292589334884454936, 3.79147058198063913458220404190, 4.83572521766157814957207479274, 5.76577287467986868605394497451, 7.04653311529225831399382324712, 8.303205770550017076020030784646, 9.555064979697189540049416641164, 10.38052610313497186183307126726, 11.36305451795361178871338182561

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