Properties

Degree 2
Conductor $ 2^{2} $
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

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-0.0680 - 0.997i$
motivic weight  =  \(36\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :18),\ -0.0680 - 0.997i)\)
\(L(\frac{37}{2})\)  \(\approx\)  \(2.074889141\)
\(L(\frac12)\)  \(\approx\)  \(2.074889141\)
\(L(19)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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