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Properties

Label 2-252-63.61-c2-0-14
Degree $2$
Conductor $252$
Sign $-0.0555 + 0.998i$
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  + (2.63 − 1.44i)3-s − 4.01i·5-s + (1.61 − 6.81i)7-s + (4.85 − 7.58i)9-s − 5.87·11-s + (−19.5 + 11.2i)13-s + (−5.78 − 10.5i)15-s + (18.7 − 10.8i)17-s + (−3.21 − 1.85i)19-s + (−5.56 − 20.2i)21-s + 4.08·23-s + 8.87·25-s + (1.84 − 26.9i)27-s + (−2.09 + 3.62i)29-s + (22.2 + 12.8i)31-s + ⋯
L(s)  = 1  + (0.877 − 0.480i)3-s − 0.803i·5-s + (0.230 − 0.973i)7-s + (0.538 − 0.842i)9-s − 0.533·11-s + (−1.50 + 0.866i)13-s + (−0.385 − 0.704i)15-s + (1.10 − 0.637i)17-s + (−0.169 − 0.0977i)19-s + (−0.265 − 0.964i)21-s + 0.177·23-s + 0.354·25-s + (0.0682 − 0.997i)27-s + (−0.0721 + 0.125i)29-s + (0.718 + 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0555 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0555 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0555 + 0.998i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ -0.0555 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.35465 - 1.43217i\)
\(L(\frac12)\) \(\approx\) \(1.35465 - 1.43217i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.63 + 1.44i)T \)
7 \( 1 + (-1.61 + 6.81i)T \)
good5 \( 1 + 4.01iT - 25T^{2} \)
11 \( 1 + 5.87T + 121T^{2} \)
13 \( 1 + (19.5 - 11.2i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-18.7 + 10.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3.21 + 1.85i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 - 4.08T + 529T^{2} \)
29 \( 1 + (2.09 - 3.62i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-22.2 - 12.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (13.5 - 23.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-47.3 + 27.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 + 20.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-61.1 + 35.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-6.84 - 11.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-53.8 - 31.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.4 - 11.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (45.1 - 78.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 2.05T + 5.04e3T^{2} \)
73 \( 1 + (62.0 - 35.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-56.9 - 98.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-132. - 76.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (123. + 71.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-21.4 - 12.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.92426379787889993840502729450, −10.38564110728939427851067260710, −9.553843681097372050149529898258, −8.614612835460018368453980493183, −7.54865194630625610998563938887, −7.00132999724354603284323190995, −5.16266403763649597297577865582, −4.13367800028497430470595722497, −2.57918467581447543093489697224, −0.978341359643658102888343647722, 2.38422221263067659194560743385, 3.13858564791528093763176665295, 4.76623834799852140949471866102, 5.85182880398034668607106622332, 7.47756035751732189215016922164, 8.054584769324934185191008647236, 9.269459351445081053663167924742, 10.13025883017159384741989120295, 10.83309538837195511683234870977, 12.20537345181850053364071022025

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