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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s + (1.5 + 0.866i)5-s + (−1.73 − 2i)7-s + (1.99 − 2i)8-s + (2.36 + 0.633i)10-s + (−0.866 + 0.5i)11-s + 3.46i·13-s + (−3.09 − 2.09i)14-s + (1.99 − 3.46i)16-s + (1.5 − 0.866i)17-s + (−2.59 + 4.5i)19-s + 3.46·20-s + (−0.999 + i)22-s + (−0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)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 + (0.748 + 0.200i)10-s + (−0.261 + 0.150i)11-s + 0.960i·13-s + (−0.827 − 0.560i)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: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16688 - 0.507617i\)
\(L(\frac12)\) \(\approx\) \(2.16688 - 0.507617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
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

−12.15157300409868079122535219568, −11.03892830183951930682650041120, −10.20142035510726389783367778720, −9.532816657576240441042287640156, −7.71833459776014426202270358411, −6.63308788477749874791497911961, −5.95553611381268526399009951696, −4.51762269512795833919696319767, −3.40830712266276344739602648842, −1.95087455775970083645575875624, 2.28880893650311205538905808938, 3.52404462442427195861886447252, 5.17880467708592840293420794750, 5.76142353580664621155347976565, 6.83928423844414598813248344312, 8.116855091635264218708372857772, 9.181348077569318399048817928203, 10.31808607236087242506603956897, 11.36587051624286443948099249136, 12.51102737175924449401166341410

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