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Properties

Label 2-252-4.3-c2-0-27
Degree $2$
Conductor $252$
Sign $0.722 + 0.691i$
Analytic cond. $6.86650$
Root an. cond. $2.62040$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.83 − 0.785i)2-s + (2.76 − 2.88i)4-s + 8.17·5-s − 2.64i·7-s + (2.81 − 7.48i)8-s + (15.0 − 6.42i)10-s + 15.5i·11-s − 20.4·13-s + (−2.07 − 4.86i)14-s + (−0.701 − 15.9i)16-s + 5.97·17-s + 4.19i·19-s + (22.6 − 23.6i)20-s + (12.2 + 28.6i)22-s − 29.3i·23-s + ⋯
L(s)  = 1  + (0.919 − 0.392i)2-s + (0.691 − 0.722i)4-s + 1.63·5-s − 0.377i·7-s + (0.352 − 0.935i)8-s + (1.50 − 0.642i)10-s + 1.41i·11-s − 1.57·13-s + (−0.148 − 0.347i)14-s + (−0.0438 − 0.999i)16-s + 0.351·17-s + 0.220i·19-s + (1.13 − 1.18i)20-s + (0.555 + 1.30i)22-s − 1.27i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ 0.722 + 0.691i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.06393 - 1.22995i\)
\(L(\frac12)\) \(\approx\) \(3.06393 - 1.22995i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.83 + 0.785i)T \)
3 \( 1 \)
7 \( 1 + 2.64iT \)
good5 \( 1 - 8.17T + 25T^{2} \)
11 \( 1 - 15.5iT - 121T^{2} \)
13 \( 1 + 20.4T + 169T^{2} \)
17 \( 1 - 5.97T + 289T^{2} \)
19 \( 1 - 4.19iT - 361T^{2} \)
23 \( 1 + 29.3iT - 529T^{2} \)
29 \( 1 - 7.89T + 841T^{2} \)
31 \( 1 - 35.3iT - 961T^{2} \)
37 \( 1 + 49.2T + 1.36e3T^{2} \)
41 \( 1 - 4.11T + 1.68e3T^{2} \)
43 \( 1 + 19.6iT - 1.84e3T^{2} \)
47 \( 1 - 30.5iT - 2.20e3T^{2} \)
53 \( 1 + 40.4T + 2.80e3T^{2} \)
59 \( 1 - 83.1iT - 3.48e3T^{2} \)
61 \( 1 + 3.43T + 3.72e3T^{2} \)
67 \( 1 - 5.17iT - 4.48e3T^{2} \)
71 \( 1 - 75.1iT - 5.04e3T^{2} \)
73 \( 1 + 21.5T + 5.32e3T^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 + 58.0iT - 6.88e3T^{2} \)
89 \( 1 - 135.T + 7.92e3T^{2} \)
97 \( 1 - 61.8T + 9.40e3T^{2} \)
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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

−12.15759297109257409928593022812, −10.36307462835332048903475528870, −10.21754760240950786992181232411, −9.250034145365656606602484700655, −7.29725769954609331194488266267, −6.53153707235630932694330255300, −5.29207035504288866703062729583, −4.57703950084335610532838597755, −2.69715197689494351950952293759, −1.72766111432829072856611742054, 2.08014835690906896639222213825, 3.20733304763936678510245947722, 5.07252529131239786191039343046, 5.68236344952853509400372102353, 6.57441874043086571169584177347, 7.82152967330396910321356740947, 9.088225108272317651196500603991, 9.995250905036845663186283358901, 11.16221582404342950263131417787, 12.14347625567671311092363223065

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