Properties

Degree $2$
Conductor $4$
Sign $-0.892 + 0.451i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.52e4 − 6.37e4i)2-s + 6.99e7i·3-s + (−3.83e9 + 1.94e9i)4-s − 1.46e11·5-s + (4.45e12 − 1.06e12i)6-s + 5.71e13i·7-s + (1.82e14 + 2.14e14i)8-s − 3.03e15·9-s + (2.23e15 + 9.34e15i)10-s + 6.39e16i·11-s + (−1.35e17 − 2.67e17i)12-s + 3.82e17·13-s + (3.64e18 − 8.70e17i)14-s − 1.02e19i·15-s + (1.09e19 − 1.48e19i)16-s − 4.07e19·17-s + ⋯
L(s)  = 1  + (−0.232 − 0.972i)2-s + 1.62i·3-s + (−0.892 + 0.451i)4-s − 0.961·5-s + (1.57 − 0.377i)6-s + 1.72i·7-s + (0.646 + 0.762i)8-s − 1.63·9-s + (0.223 + 0.934i)10-s + 1.39i·11-s + (−0.733 − 1.44i)12-s + 0.574·13-s + (1.67 − 0.399i)14-s − 1.56i·15-s + (0.591 − 0.806i)16-s − 0.837·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.892 + 0.451i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.6833074930\)
\(L(\frac12)\) \(\approx\) \(0.6833074930\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.52e4 + 6.37e4i)T \)
good3 \( 1 - 6.99e7iT - 1.85e15T^{2} \)
5 \( 1 + 1.46e11T + 2.32e22T^{2} \)
7 \( 1 - 5.71e13iT - 1.10e27T^{2} \)
11 \( 1 - 6.39e16iT - 2.11e33T^{2} \)
13 \( 1 - 3.82e17T + 4.42e35T^{2} \)
17 \( 1 + 4.07e19T + 2.36e39T^{2} \)
19 \( 1 + 6.13e19iT - 8.31e40T^{2} \)
23 \( 1 + 2.81e20iT - 3.76e43T^{2} \)
29 \( 1 - 9.24e22T + 6.26e46T^{2} \)
31 \( 1 - 4.23e23iT - 5.29e47T^{2} \)
37 \( 1 + 4.96e23T + 1.52e50T^{2} \)
41 \( 1 - 7.78e25T + 4.06e51T^{2} \)
43 \( 1 - 5.67e24iT - 1.86e52T^{2} \)
47 \( 1 + 2.70e26iT - 3.21e53T^{2} \)
53 \( 1 - 1.23e27T + 1.50e55T^{2} \)
59 \( 1 + 3.43e27iT - 4.64e56T^{2} \)
61 \( 1 - 4.92e28T + 1.35e57T^{2} \)
67 \( 1 - 1.86e29iT - 2.71e58T^{2} \)
71 \( 1 + 7.39e29iT - 1.73e59T^{2} \)
73 \( 1 - 1.01e30T + 4.22e59T^{2} \)
79 \( 1 - 2.63e30iT - 5.29e60T^{2} \)
83 \( 1 - 5.56e30iT - 2.57e61T^{2} \)
89 \( 1 - 1.36e31T + 2.40e62T^{2} \)
97 \( 1 + 9.37e31T + 3.77e63T^{2} \)
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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

−17.93694737542076059508087521050, −15.86697467472792754229819686330, −15.00527860695863652128409944625, −12.28251731241257821486258525797, −11.13904515185557166172191475630, −9.586966661611920501660562857574, −8.556198466636276862970903620399, −5.02035783725721725390619618172, −3.91476788377801770756115578943, −2.43750937467631057657562525146, 0.31902615374232653937731624382, 0.991190937236505938020415310219, 3.89282474232847998717877210938, 6.30695703575476155786899823395, 7.42226052487948717552396843433, 8.290827934663313274010594840215, 11.09613909222827471163569703726, 13.20828309805895705368932460508, 14.01466868335186720086198847632, 16.20011391145279698403496147570

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