Properties

Degree $2$
Conductor $4$
Sign $0.287 + 0.957i$
Motivic weight $36$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.10e5 − 1.56e5i)2-s + 7.14e8i·3-s + (1.97e10 + 6.58e10i)4-s + 8.84e11·5-s + (1.11e14 − 1.50e14i)6-s + 1.08e15i·7-s + (6.14e15 − 1.69e16i)8-s − 3.60e17·9-s + (−1.86e17 − 1.38e17i)10-s − 6.19e18i·11-s + (−4.70e19 + 1.41e19i)12-s − 1.29e20·13-s + (1.69e20 − 2.27e20i)14-s + 6.31e20i·15-s + (−3.94e21 + 2.60e21i)16-s + 1.60e22·17-s + ⋯
L(s)  = 1  + (−0.802 − 0.596i)2-s + 1.84i·3-s + (0.287 + 0.957i)4-s + 0.231·5-s + (1.10 − 1.47i)6-s + 0.664i·7-s + (0.340 − 0.940i)8-s − 2.39·9-s + (−0.186 − 0.138i)10-s − 1.11i·11-s + (−1.76 + 0.530i)12-s − 1.15·13-s + (0.396 − 0.533i)14-s + 0.427i·15-s + (−0.834 + 0.550i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.287 + 0.957i$
Motivic weight: \(36\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :18),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(0.3243634107\)
\(L(\frac12)\) \(\approx\) \(0.3243634107\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.10e5 + 1.56e5i)T \)
good3 \( 1 - 7.14e8iT - 1.50e17T^{2} \)
5 \( 1 - 8.84e11T + 1.45e25T^{2} \)
7 \( 1 - 1.08e15iT - 2.65e30T^{2} \)
11 \( 1 + 6.19e18iT - 3.09e37T^{2} \)
13 \( 1 + 1.29e20T + 1.26e40T^{2} \)
17 \( 1 - 1.60e22T + 1.97e44T^{2} \)
19 \( 1 - 7.21e22iT - 1.08e46T^{2} \)
23 \( 1 + 5.50e24iT - 1.05e49T^{2} \)
29 \( 1 + 2.40e26T + 4.42e52T^{2} \)
31 \( 1 - 3.47e26iT - 4.88e53T^{2} \)
37 \( 1 + 4.85e26T + 2.85e56T^{2} \)
41 \( 1 - 1.61e28T + 1.14e58T^{2} \)
43 \( 1 + 2.17e29iT - 6.38e58T^{2} \)
47 \( 1 - 3.68e28iT - 1.56e60T^{2} \)
53 \( 1 - 5.88e30T + 1.18e62T^{2} \)
59 \( 1 + 7.36e31iT - 5.63e63T^{2} \)
61 \( 1 - 2.36e32T + 1.87e64T^{2} \)
67 \( 1 - 3.51e32iT - 5.47e65T^{2} \)
71 \( 1 + 7.98e32iT - 4.41e66T^{2} \)
73 \( 1 + 4.73e33T + 1.20e67T^{2} \)
79 \( 1 - 8.69e32iT - 2.06e68T^{2} \)
83 \( 1 - 2.72e34iT - 1.22e69T^{2} \)
89 \( 1 - 4.88e34T + 1.50e70T^{2} \)
97 \( 1 + 1.09e35T + 3.34e71T^{2} \)
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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

−16.15128728778875523138400051085, −14.58490769584275162888083578195, −11.96040172663807967984707836819, −10.54560900301939100408536624379, −9.578898518586597763230288006330, −8.398553524681152234146070308884, −5.52133235848995233961683521904, −3.79768131585325618287276196837, −2.58291757322873207206874775784, −0.13776279888503047230612645353, 1.16519279409216299872931555142, 2.20204161769867009707526607515, 5.59424365591197862083062447961, 7.20684570633506588117007179847, 7.61622391619597805194008378312, 9.664062255483504211476857456084, 11.75840879048722980967561682148, 13.35933256935609345759381856840, 14.70010925049958174673476765819, 17.12216682914165661394751084513

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