Properties

Degree $2$
Conductor $4$
Sign $0.755 + 0.655i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.14e4 − 2.29e4i)2-s + 4.43e7i·3-s + (3.24e9 + 2.81e9i)4-s + 1.39e11·5-s + (1.01e12 − 2.72e12i)6-s − 2.23e13i·7-s + (−1.34e14 − 2.47e14i)8-s − 1.15e14·9-s + (−8.54e15 − 3.18e15i)10-s − 1.78e16i·11-s + (−1.24e17 + 1.43e17i)12-s + 2.16e17·13-s + (−5.11e17 + 1.37e18i)14-s + 6.17e18i·15-s + (2.61e18 + 1.82e19i)16-s − 6.14e19·17-s + ⋯
L(s)  = 1  + (−0.936 − 0.349i)2-s + 1.03i·3-s + (0.755 + 0.655i)4-s + 0.912·5-s + (0.360 − 0.965i)6-s − 0.672i·7-s + (−0.478 − 0.877i)8-s − 0.0623·9-s + (−0.854 − 0.318i)10-s − 0.389i·11-s + (−0.675 + 0.778i)12-s + 0.325·13-s + (−0.235 + 0.629i)14-s + 0.940i·15-s + (0.141 + 0.989i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.755 + 0.655i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ 0.755 + 0.655i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.375112421\)
\(L(\frac12)\) \(\approx\) \(1.375112421\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.14e4 + 2.29e4i)T \)
good3 \( 1 - 4.43e7iT - 1.85e15T^{2} \)
5 \( 1 - 1.39e11T + 2.32e22T^{2} \)
7 \( 1 + 2.23e13iT - 1.10e27T^{2} \)
11 \( 1 + 1.78e16iT - 2.11e33T^{2} \)
13 \( 1 - 2.16e17T + 4.42e35T^{2} \)
17 \( 1 + 6.14e19T + 2.36e39T^{2} \)
19 \( 1 + 4.44e20iT - 8.31e40T^{2} \)
23 \( 1 + 8.74e21iT - 3.76e43T^{2} \)
29 \( 1 - 4.53e23T + 6.26e46T^{2} \)
31 \( 1 + 1.07e24iT - 5.29e47T^{2} \)
37 \( 1 + 9.58e23T + 1.52e50T^{2} \)
41 \( 1 + 1.42e25T + 4.06e51T^{2} \)
43 \( 1 - 1.36e26iT - 1.86e52T^{2} \)
47 \( 1 + 1.63e26iT - 3.21e53T^{2} \)
53 \( 1 - 6.59e27T + 1.50e55T^{2} \)
59 \( 1 - 2.05e26iT - 4.64e56T^{2} \)
61 \( 1 - 1.66e28T + 1.35e57T^{2} \)
67 \( 1 + 2.31e29iT - 2.71e58T^{2} \)
71 \( 1 - 3.68e29iT - 1.73e59T^{2} \)
73 \( 1 - 2.79e29T + 4.22e59T^{2} \)
79 \( 1 + 2.19e30iT - 5.29e60T^{2} \)
83 \( 1 - 3.91e30iT - 2.57e61T^{2} \)
89 \( 1 + 2.76e31T + 2.40e62T^{2} \)
97 \( 1 - 2.89e30T + 3.77e63T^{2} \)
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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

−16.81608746611385388807454326968, −15.54197665567965471081505911932, −13.36973821403625393190237935029, −11.02434028292786092435359036495, −10.02937510399076036663637165710, −8.807286009869433778354670440536, −6.62721960008975960967056576788, −4.32340190530278279496337715737, −2.49280493029866675264390903314, −0.65727447862748033665491306779, 1.32574118559947280449017266610, 2.19917284530657578357194319583, 5.76051166789296085697664462658, 6.94073657568444553624400773392, 8.552326881848522267617333126687, 10.09163319222092994240070628247, 12.10197142163130178914594355823, 13.81451990529279830590962740523, 15.66712305886866827198327348388, 17.57191090773405555766521575922

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