Properties

Degree $2$
Conductor $4$
Sign $-0.522 - 0.852i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.20e4 − 5.71e4i)2-s + 1.43e7i·3-s + (−2.24e9 − 3.66e9i)4-s − 4.45e10·5-s + (8.19e11 + 4.58e11i)6-s − 4.42e13i·7-s + (−2.81e14 + 1.12e13i)8-s + 1.64e15·9-s + (−1.42e15 + 2.54e15i)10-s + 2.28e16i·11-s + (5.24e16 − 3.21e16i)12-s − 7.09e17·13-s + (−2.53e18 − 1.41e18i)14-s − 6.38e17i·15-s + (−8.36e18 + 1.64e19i)16-s − 5.13e19·17-s + ⋯
L(s)  = 1  + (0.488 − 0.872i)2-s + 0.332i·3-s + (−0.522 − 0.852i)4-s − 0.292·5-s + (0.290 + 0.162i)6-s − 1.33i·7-s + (−0.999 + 0.0398i)8-s + 0.889·9-s + (−0.142 + 0.254i)10-s + 0.497i·11-s + (0.283 − 0.174i)12-s − 1.06·13-s + (−1.16 − 0.650i)14-s − 0.0972i·15-s + (−0.453 + 0.891i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.522 - 0.852i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ -0.522 - 0.852i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.3684140492\)
\(L(\frac12)\) \(\approx\) \(0.3684140492\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.20e4 + 5.71e4i)T \)
good3 \( 1 - 1.43e7iT - 1.85e15T^{2} \)
5 \( 1 + 4.45e10T + 2.32e22T^{2} \)
7 \( 1 + 4.42e13iT - 1.10e27T^{2} \)
11 \( 1 - 2.28e16iT - 2.11e33T^{2} \)
13 \( 1 + 7.09e17T + 4.42e35T^{2} \)
17 \( 1 + 5.13e19T + 2.36e39T^{2} \)
19 \( 1 - 9.08e19iT - 8.31e40T^{2} \)
23 \( 1 - 3.40e21iT - 3.76e43T^{2} \)
29 \( 1 + 9.75e22T + 6.26e46T^{2} \)
31 \( 1 - 8.90e23iT - 5.29e47T^{2} \)
37 \( 1 + 1.36e25T + 1.52e50T^{2} \)
41 \( 1 - 9.45e25T + 4.06e51T^{2} \)
43 \( 1 + 2.05e26iT - 1.86e52T^{2} \)
47 \( 1 + 6.70e25iT - 3.21e53T^{2} \)
53 \( 1 + 3.12e27T + 1.50e55T^{2} \)
59 \( 1 + 4.03e28iT - 4.64e56T^{2} \)
61 \( 1 + 4.61e28T + 1.35e57T^{2} \)
67 \( 1 + 2.83e29iT - 2.71e58T^{2} \)
71 \( 1 - 3.78e29iT - 1.73e59T^{2} \)
73 \( 1 + 8.28e29T + 4.22e59T^{2} \)
79 \( 1 - 3.22e30iT - 5.29e60T^{2} \)
83 \( 1 - 3.96e29iT - 2.57e61T^{2} \)
89 \( 1 - 1.44e31T + 2.40e62T^{2} \)
97 \( 1 - 5.69e30T + 3.77e63T^{2} \)
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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

−15.51448269627413244007352333110, −13.92757792407982489094629824349, −12.50423595156291650410879589728, −10.76901329606727624879236541991, −9.679363736347633408081995486239, −7.16408161032747303087758809670, −4.73726168667176252870942628342, −3.75608466357675924408105734550, −1.73766525367221020102943367279, −0.098759474866183931409339251022, 2.48502706778180948986390503951, 4.48174841583370276438922541197, 6.07113246530241338367473689295, 7.59448601685614221114195287210, 9.140334657735479148115134412393, 11.92401247100302308911426330110, 13.12082649214843069216337643384, 14.93881869860480484744171579604, 15.96706766780928901889923361941, 17.77574858149185229453587528543

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