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Properties

Label 2-252-28.27-c3-0-16
Degree $2$
Conductor $252$
Sign $0.341 - 0.939i$
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  + (2.41 − 1.47i)2-s + (3.65 − 7.11i)4-s + 17.0i·5-s + (−18.3 + 2.33i)7-s + (−1.65 − 22.5i)8-s + (25.1 + 41.2i)10-s + 41.4i·11-s + 45.3i·13-s + (−40.9 + 32.7i)14-s + (−37.2 − 52.0i)16-s + 28.2i·17-s + 41.5·19-s + (121. + 62.4i)20-s + (61.1 + 100. i)22-s + 93.9i·23-s + ⋯
L(s)  = 1  + (0.853 − 0.521i)2-s + (0.457 − 0.889i)4-s + 1.52i·5-s + (−0.992 + 0.126i)7-s + (−0.0732 − 0.997i)8-s + (0.795 + 1.30i)10-s + 1.13i·11-s + 0.967i·13-s + (−0.780 + 0.624i)14-s + (−0.582 − 0.813i)16-s + 0.403i·17-s + 0.501·19-s + (1.35 + 0.698i)20-s + (0.592 + 0.970i)22-s + 0.851i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.341 - 0.939i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.341 - 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.154836087\)
\(L(\frac12)\) \(\approx\) \(2.154836087\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.41 + 1.47i)T \)
3 \( 1 \)
7 \( 1 + (18.3 - 2.33i)T \)
good5 \( 1 - 17.0iT - 125T^{2} \)
11 \( 1 - 41.4iT - 1.33e3T^{2} \)
13 \( 1 - 45.3iT - 2.19e3T^{2} \)
17 \( 1 - 28.2iT - 4.91e3T^{2} \)
19 \( 1 - 41.5T + 6.85e3T^{2} \)
23 \( 1 - 93.9iT - 1.21e4T^{2} \)
29 \( 1 + 27.8T + 2.43e4T^{2} \)
31 \( 1 + 81.4T + 2.97e4T^{2} \)
37 \( 1 - 94.8T + 5.06e4T^{2} \)
41 \( 1 - 227. iT - 6.89e4T^{2} \)
43 \( 1 + 171. iT - 7.95e4T^{2} \)
47 \( 1 - 286.T + 1.03e5T^{2} \)
53 \( 1 - 575.T + 1.48e5T^{2} \)
59 \( 1 + 411.T + 2.05e5T^{2} \)
61 \( 1 + 778. iT - 2.26e5T^{2} \)
67 \( 1 + 198. iT - 3.00e5T^{2} \)
71 \( 1 - 197. iT - 3.57e5T^{2} \)
73 \( 1 - 255. iT - 3.89e5T^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + 938.T + 5.71e5T^{2} \)
89 \( 1 + 1.16e3iT - 7.04e5T^{2} \)
97 \( 1 + 656. iT - 9.12e5T^{2} \)
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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

−11.79220627213197026556989029494, −10.95733243596848891308793373560, −10.01142380082412299905300937239, −9.442973590524492901061154400984, −7.26116456120390643294755531906, −6.74840318352915218401253084135, −5.73013997566860351274435955430, −4.13412141401664839718835012810, −3.14548395090374850241974684114, −2.03946308357283813317215210957, 0.62411572148362948113658779062, 2.93685253427161721932589385237, 4.11532167147719291198302511867, 5.35144037972595048008750773160, 5.99975595976728735892270207600, 7.38383415309115739824430011828, 8.456564542308625503732523205971, 9.160666857516900456634597929994, 10.56787906396256647428207420984, 11.85128120716681465962554135198

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