Properties

Degree $2$
Conductor $4$
Sign $-0.0106 - 0.999i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.60e4 − 4.65e4i)2-s − 8.00e7i·3-s + (−4.59e7 − 4.29e9i)4-s − 3.25e10·5-s + (−3.73e12 − 3.69e12i)6-s + 4.74e13i·7-s + (−2.02e14 − 1.95e14i)8-s − 4.55e15·9-s + (−1.49e15 + 1.51e15i)10-s + 1.35e16i·11-s + (−3.43e17 + 3.67e15i)12-s + 9.45e16·13-s + (2.21e18 + 2.18e18i)14-s + 2.60e18i·15-s + (−1.84e19 + 3.94e17i)16-s + 6.35e18·17-s + ⋯
L(s)  = 1  + (0.703 − 0.710i)2-s − 1.85i·3-s + (−0.0106 − 0.999i)4-s − 0.213·5-s + (−1.32 − 1.30i)6-s + 1.42i·7-s + (−0.718 − 0.695i)8-s − 2.45·9-s + (−0.149 + 0.151i)10-s + 0.294i·11-s + (−1.85 + 0.0198i)12-s + 0.142·13-s + (1.01 + 1.00i)14-s + 0.396i·15-s + (−0.999 + 0.0213i)16-s + 0.130·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.0106 - 0.999i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ -0.0106 - 0.999i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.126043129\)
\(L(\frac12)\) \(\approx\) \(1.126043129\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.60e4 + 4.65e4i)T \)
good3 \( 1 + 8.00e7iT - 1.85e15T^{2} \)
5 \( 1 + 3.25e10T + 2.32e22T^{2} \)
7 \( 1 - 4.74e13iT - 1.10e27T^{2} \)
11 \( 1 - 1.35e16iT - 2.11e33T^{2} \)
13 \( 1 - 9.45e16T + 4.42e35T^{2} \)
17 \( 1 - 6.35e18T + 2.36e39T^{2} \)
19 \( 1 + 4.93e20iT - 8.31e40T^{2} \)
23 \( 1 + 4.92e21iT - 3.76e43T^{2} \)
29 \( 1 + 2.04e23T + 6.26e46T^{2} \)
31 \( 1 - 8.15e23iT - 5.29e47T^{2} \)
37 \( 1 - 8.64e24T + 1.52e50T^{2} \)
41 \( 1 + 9.37e25T + 4.06e51T^{2} \)
43 \( 1 + 9.13e25iT - 1.86e52T^{2} \)
47 \( 1 + 4.69e26iT - 3.21e53T^{2} \)
53 \( 1 - 3.40e27T + 1.50e55T^{2} \)
59 \( 1 - 2.87e27iT - 4.64e56T^{2} \)
61 \( 1 - 7.72e27T + 1.35e57T^{2} \)
67 \( 1 + 8.28e28iT - 2.71e58T^{2} \)
71 \( 1 - 2.37e29iT - 1.73e59T^{2} \)
73 \( 1 + 4.61e27T + 4.22e59T^{2} \)
79 \( 1 + 3.06e30iT - 5.29e60T^{2} \)
83 \( 1 - 3.67e29iT - 2.57e61T^{2} \)
89 \( 1 + 2.12e31T + 2.40e62T^{2} \)
97 \( 1 - 2.45e30T + 3.77e63T^{2} \)
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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

−15.09684352795566957125222549284, −13.50480760027237797194152525133, −12.39612133969084622235574072687, −11.53009699565604306206571639801, −8.762066383304908450583588227775, −6.78684903504018693892475651472, −5.45413145572690612875919848553, −2.75936959394890153380692633726, −1.82302346163115416272555654209, −0.27698848101185851346731347330, 3.56118118734662589005560453815, 4.16085331745688948256740230658, 5.70550035631374908790399072473, 7.941219183433904717347664172650, 9.845163616208985422195770901012, 11.32727507847606811427552799316, 13.80178996427195333607106270202, 14.99408865330907150469509636271, 16.31701016894955683181246854640, 17.02632053762628560939976985435

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