Properties

Degree $2$
Conductor $252$
Sign $0.895 - 0.444i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + (1.5 − 0.866i)5-s + (1.73 − 2i)7-s + (2 − 1.99i)8-s + (0.633 + 2.36i)10-s + (0.866 + 0.5i)11-s − 3.46i·13-s + (2.09 + 3.09i)14-s + (1.99 + 3.46i)16-s + (1.5 + 0.866i)17-s + (2.59 + 4.5i)19-s − 3.46·20-s + (−1 + 0.999i)22-s + (0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.670 − 0.387i)5-s + (0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + (0.200 + 0.748i)10-s + (0.261 + 0.150i)11-s − 0.960i·13-s + (0.560 + 0.827i)14-s + (0.499 + 0.866i)16-s + (0.363 + 0.210i)17-s + (0.596 + 1.03i)19-s − 0.774·20-s + (−0.213 + 0.213i)22-s + (0.180 − 0.104i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.895 - 0.444i$
Motivic weight: \(1\)
Character: $\chi_{252} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17801 + 0.275962i\)
\(L(\frac12)\) \(\approx\) \(1.17801 + 0.275962i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-0.866 + 1.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (-4.33 - 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−12.34644501914327901913710598803, −10.83330456132843664996399786308, −10.03063330434424302755149810261, −9.153773890876115466387162209172, −8.001127700191252488468349354212, −7.35453885307732788569458248129, −5.94276809123589883637803203014, −5.20090128908411526690049517406, −3.88970669704056830956576049036, −1.32934057357336482862740897306, 1.74866543200013856322997759698, 2.93246459607072519749466339585, 4.52740840318411717384902245114, 5.65455300907543654002181839811, 7.10986310140454186452835465604, 8.443681402675250788071282963638, 9.258587607965775873298110239317, 10.02133305920299209255437259453, 11.26336100624428210761731770021, 11.65966563460862894801018543287

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