Properties

Degree $2$
Conductor $4$
Sign $0.461 + 0.887i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.60e4 − 3.40e4i)2-s − 4.09e7i·3-s + (1.98e9 + 3.81e9i)4-s − 2.17e11·5-s + (−1.39e12 + 2.29e12i)6-s − 2.81e12i·7-s + (1.85e13 − 2.80e14i)8-s + 1.79e14·9-s + (1.21e16 + 7.38e15i)10-s + 6.95e16i·11-s + (1.55e17 − 8.10e16i)12-s − 7.08e17·13-s + (−9.58e16 + 1.57e17i)14-s + 8.88e18i·15-s + (−1.05e19 + 1.51e19i)16-s + 6.55e19·17-s + ⋯
L(s)  = 1  + (−0.854 − 0.518i)2-s − 0.950i·3-s + (0.461 + 0.887i)4-s − 1.42·5-s + (−0.493 + 0.812i)6-s − 0.0847i·7-s + (0.0658 − 0.997i)8-s + 0.0970·9-s + (1.21 + 0.738i)10-s + 1.51i·11-s + (0.843 − 0.438i)12-s − 1.06·13-s + (−0.0440 + 0.0724i)14-s + 1.35i·15-s + (−0.574 + 0.818i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.461 + 0.887i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ 0.461 + 0.887i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.7985814603\)
\(L(\frac12)\) \(\approx\) \(0.7985814603\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.60e4 + 3.40e4i)T \)
good3 \( 1 + 4.09e7iT - 1.85e15T^{2} \)
5 \( 1 + 2.17e11T + 2.32e22T^{2} \)
7 \( 1 + 2.81e12iT - 1.10e27T^{2} \)
11 \( 1 - 6.95e16iT - 2.11e33T^{2} \)
13 \( 1 + 7.08e17T + 4.42e35T^{2} \)
17 \( 1 - 6.55e19T + 2.36e39T^{2} \)
19 \( 1 - 1.83e20iT - 8.31e40T^{2} \)
23 \( 1 + 7.92e20iT - 3.76e43T^{2} \)
29 \( 1 + 1.45e23T + 6.26e46T^{2} \)
31 \( 1 + 1.34e24iT - 5.29e47T^{2} \)
37 \( 1 + 2.62e24T + 1.52e50T^{2} \)
41 \( 1 + 1.80e25T + 4.06e51T^{2} \)
43 \( 1 + 1.56e26iT - 1.86e52T^{2} \)
47 \( 1 - 8.64e26iT - 3.21e53T^{2} \)
53 \( 1 - 5.50e27T + 1.50e55T^{2} \)
59 \( 1 - 9.48e27iT - 4.64e56T^{2} \)
61 \( 1 - 3.04e28T + 1.35e57T^{2} \)
67 \( 1 + 7.40e28iT - 2.71e58T^{2} \)
71 \( 1 + 2.60e29iT - 1.73e59T^{2} \)
73 \( 1 - 3.70e29T + 4.22e59T^{2} \)
79 \( 1 - 2.10e30iT - 5.29e60T^{2} \)
83 \( 1 - 6.24e30iT - 2.57e61T^{2} \)
89 \( 1 - 1.03e31T + 2.40e62T^{2} \)
97 \( 1 - 1.05e32T + 3.77e63T^{2} \)
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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

−16.93334212134248685275676087200, −15.20272381146639860007558662187, −12.53035700233936447038798169847, −11.93278182955552465269260397329, −9.932504251427841540191471674968, −7.74656458696902460072253560896, −7.30154820687368695740601827303, −4.02353646193843535569200453066, −2.13755417494897341293527274274, −0.65440036514173282534173907012, 0.65475533794856822958827552831, 3.39142768925149755064810627838, 5.15118526722958919184489368977, 7.34725893139965725663788911693, 8.707236888345447629500564188533, 10.32168520648669381356187652769, 11.66566726480471700217776534146, 14.67424333093998251636211980887, 15.83562353638167873353992116228, 16.64069403058116223280417464672

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