Properties

Degree $2$
Conductor $4$
Sign $-0.848 + 0.529i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80e4 − 6.29e4i)2-s − 2.85e7i·3-s + (−3.64e9 + 2.27e9i)4-s + 1.21e11·5-s + (−1.79e12 + 5.15e11i)6-s + 3.17e10i·7-s + (2.09e14 + 1.88e14i)8-s + 1.03e15·9-s + (−2.18e15 − 7.63e15i)10-s − 6.68e16i·11-s + (6.49e16 + 1.04e17i)12-s + 5.92e17·13-s + (1.99e15 − 5.72e14i)14-s − 3.46e18i·15-s + (8.09e18 − 1.65e19i)16-s + 4.82e19·17-s + ⋯
L(s)  = 1  + (−0.275 − 0.961i)2-s − 0.663i·3-s + (−0.848 + 0.529i)4-s + 0.794·5-s + (−0.637 + 0.182i)6-s + 0.000954i·7-s + (0.742 + 0.669i)8-s + 0.560·9-s + (−0.218 − 0.763i)10-s − 1.45i·11-s + (0.351 + 0.562i)12-s + 0.889·13-s + (0.000917 − 0.000263i)14-s − 0.527i·15-s + (0.438 − 0.898i)16-s + 0.992·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.848 + 0.529i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ -0.848 + 0.529i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.873175936\)
\(L(\frac12)\) \(\approx\) \(1.873175936\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.80e4 + 6.29e4i)T \)
good3 \( 1 + 2.85e7iT - 1.85e15T^{2} \)
5 \( 1 - 1.21e11T + 2.32e22T^{2} \)
7 \( 1 - 3.17e10iT - 1.10e27T^{2} \)
11 \( 1 + 6.68e16iT - 2.11e33T^{2} \)
13 \( 1 - 5.92e17T + 4.42e35T^{2} \)
17 \( 1 - 4.82e19T + 2.36e39T^{2} \)
19 \( 1 - 9.57e19iT - 8.31e40T^{2} \)
23 \( 1 - 5.33e20iT - 3.76e43T^{2} \)
29 \( 1 + 4.06e23T + 6.26e46T^{2} \)
31 \( 1 + 4.12e23iT - 5.29e47T^{2} \)
37 \( 1 - 1.13e25T + 1.52e50T^{2} \)
41 \( 1 + 2.76e23T + 4.06e51T^{2} \)
43 \( 1 - 1.68e26iT - 1.86e52T^{2} \)
47 \( 1 + 7.88e26iT - 3.21e53T^{2} \)
53 \( 1 + 5.53e27T + 1.50e55T^{2} \)
59 \( 1 + 4.25e28iT - 4.64e56T^{2} \)
61 \( 1 - 5.00e28T + 1.35e57T^{2} \)
67 \( 1 + 1.33e29iT - 2.71e58T^{2} \)
71 \( 1 + 5.68e29iT - 1.73e59T^{2} \)
73 \( 1 + 6.17e29T + 4.22e59T^{2} \)
79 \( 1 - 1.44e30iT - 5.29e60T^{2} \)
83 \( 1 - 6.98e30iT - 2.57e61T^{2} \)
89 \( 1 + 1.07e31T + 2.40e62T^{2} \)
97 \( 1 + 3.35e31T + 3.77e63T^{2} \)
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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

−16.66298518065404434580009458101, −13.86673339205436010341542089357, −12.94068243862405459019496327210, −11.24237490140745053763522961381, −9.675346237006784787056942138727, −8.029014640081064982199175830773, −5.84386703188054812712469366219, −3.55178030838392514253090057749, −1.84309669527337363891195207279, −0.76509847696168848584030493330, 1.47761061364304727613233228297, 4.18931983099131802872975908667, 5.62970907954782713408786354100, 7.30201709619688906842804123162, 9.303050522655654542413787466191, 10.24384391330360763272008284356, 13.06597673326915306984539789681, 14.68080467211539353380092859893, 15.89893385080015845344686342428, 17.31356498553562152610933881694

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