Properties

Degree $2$
Conductor $4$
Sign $-0.0680 + 0.997i$
Motivic weight $36$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.78e5 + 1.91e5i)2-s − 4.73e8i·3-s + (−4.67e9 + 6.85e10i)4-s + 4.97e11·5-s + (9.07e13 − 8.47e13i)6-s − 9.39e14i·7-s + (−1.39e16 + 1.13e16i)8-s − 7.41e16·9-s + (8.91e16 + 9.53e16i)10-s − 1.64e18i·11-s + (3.24e19 + 2.21e18i)12-s + 1.14e20·13-s + (1.79e20 − 1.68e20i)14-s − 2.35e20i·15-s + (−4.67e21 − 6.40e20i)16-s − 2.29e22·17-s + ⋯
L(s)  = 1  + (0.682 + 0.730i)2-s − 1.22i·3-s + (−0.0680 + 0.997i)4-s + 0.130·5-s + (0.893 − 0.834i)6-s − 0.576i·7-s + (−0.775 + 0.631i)8-s − 0.493·9-s + (0.0891 + 0.0953i)10-s − 0.296i·11-s + (1.21 + 0.0831i)12-s + 1.01·13-s + (0.421 − 0.393i)14-s − 0.159i·15-s + (−0.990 − 0.135i)16-s − 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0680 + 0.997i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.0680 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.0680 + 0.997i$
Motivic weight: \(36\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :18),\ -0.0680 + 0.997i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(2.074889141\)
\(L(\frac12)\) \(\approx\) \(2.074889141\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78e5 - 1.91e5i)T \)
good3 \( 1 + 4.73e8iT - 1.50e17T^{2} \)
5 \( 1 - 4.97e11T + 1.45e25T^{2} \)
7 \( 1 + 9.39e14iT - 2.65e30T^{2} \)
11 \( 1 + 1.64e18iT - 3.09e37T^{2} \)
13 \( 1 - 1.14e20T + 1.26e40T^{2} \)
17 \( 1 + 2.29e22T + 1.97e44T^{2} \)
19 \( 1 + 1.20e23iT - 1.08e46T^{2} \)
23 \( 1 + 2.81e24iT - 1.05e49T^{2} \)
29 \( 1 + 1.31e25T + 4.42e52T^{2} \)
31 \( 1 + 7.15e26iT - 4.88e53T^{2} \)
37 \( 1 - 2.59e28T + 2.85e56T^{2} \)
41 \( 1 + 1.71e29T + 1.14e58T^{2} \)
43 \( 1 + 4.11e29iT - 6.38e58T^{2} \)
47 \( 1 - 8.93e29iT - 1.56e60T^{2} \)
53 \( 1 - 3.45e30T + 1.18e62T^{2} \)
59 \( 1 + 2.87e31iT - 5.63e63T^{2} \)
61 \( 1 + 1.57e31T + 1.87e64T^{2} \)
67 \( 1 - 2.63e32iT - 5.47e65T^{2} \)
71 \( 1 - 2.24e33iT - 4.41e66T^{2} \)
73 \( 1 + 4.63e33T + 1.20e67T^{2} \)
79 \( 1 - 2.60e34iT - 2.06e68T^{2} \)
83 \( 1 + 9.58e32iT - 1.22e69T^{2} \)
89 \( 1 - 3.91e34T + 1.50e70T^{2} \)
97 \( 1 - 5.91e35T + 3.34e71T^{2} \)
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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

−15.57257944064008511228377894936, −13.65632737178639552993332884961, −13.13448194744606214308155736952, −11.39514022849714736022502592796, −8.547675585576572735028714414840, −7.13134853837056084317708305356, −6.17236629130166869775678384936, −4.20303766510361285462035898519, −2.30507337557090686801408920116, −0.48137984665521758223199382163, 1.76197976849507879358279702937, 3.46004559990365658323095286640, 4.59421852635989326183964074791, 6.00476956818728931547420458469, 9.047608065673765299619851056888, 10.27808654615063687375666178641, 11.55399277388843515307226600400, 13.32778972084757858677885222981, 14.99911240420177504895623076789, 15.93374279823438545505586092202

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