Properties

Degree 2
Conductor $ 2^{2} $
Sign $0.508 + 0.860i$
Motivic weight 36
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.27e5 − 1.29e5i)2-s − 1.77e8i·3-s + (3.49e10 + 5.91e10i)4-s + 5.55e12·5-s + (−2.30e13 + 4.03e13i)6-s − 2.49e15i·7-s + (−2.69e14 − 1.80e16i)8-s + 1.18e17·9-s + (−1.26e18 − 7.21e17i)10-s + 5.40e18i·11-s + (1.04e19 − 6.19e18i)12-s + 1.72e20·13-s + (−3.24e20 + 5.69e20i)14-s − 9.84e20i·15-s + (−2.27e21 + 4.13e21i)16-s + 8.60e21·17-s + ⋯
L(s)  = 1  + (−0.868 − 0.495i)2-s − 0.457i·3-s + (0.508 + 0.860i)4-s + 1.45·5-s + (−0.226 + 0.397i)6-s − 1.53i·7-s + (−0.0149 − 0.999i)8-s + 0.790·9-s + (−1.26 − 0.721i)10-s + 0.972i·11-s + (0.393 − 0.232i)12-s + 1.53·13-s + (−0.760 + 1.33i)14-s − 0.666i·15-s + (−0.482 + 0.875i)16-s + 0.611·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $0.508 + 0.860i$
motivic weight  =  \(36\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :18),\ 0.508 + 0.860i)\)
\(L(\frac{37}{2})\)  \(\approx\)  \(2.121132357\)
\(L(\frac12)\)  \(\approx\)  \(2.121132357\)
\(L(19)\)   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 \( 1 + (2.27e5 + 1.29e5i)T \)
good3 \( 1 + 1.77e8iT - 1.50e17T^{2} \)
5 \( 1 - 5.55e12T + 1.45e25T^{2} \)
7 \( 1 + 2.49e15iT - 2.65e30T^{2} \)
11 \( 1 - 5.40e18iT - 3.09e37T^{2} \)
13 \( 1 - 1.72e20T + 1.26e40T^{2} \)
17 \( 1 - 8.60e21T + 1.97e44T^{2} \)
19 \( 1 - 1.88e23iT - 1.08e46T^{2} \)
23 \( 1 - 9.41e23iT - 1.05e49T^{2} \)
29 \( 1 + 4.55e25T + 4.42e52T^{2} \)
31 \( 1 - 5.01e26iT - 4.88e53T^{2} \)
37 \( 1 - 1.31e28T + 2.85e56T^{2} \)
41 \( 1 - 2.91e27T + 1.14e58T^{2} \)
43 \( 1 + 6.20e28iT - 6.38e58T^{2} \)
47 \( 1 - 4.91e29iT - 1.56e60T^{2} \)
53 \( 1 - 4.35e29T + 1.18e62T^{2} \)
59 \( 1 + 5.43e31iT - 5.63e63T^{2} \)
61 \( 1 + 3.68e31T + 1.87e64T^{2} \)
67 \( 1 + 9.33e32iT - 5.47e65T^{2} \)
71 \( 1 - 1.44e33iT - 4.41e66T^{2} \)
73 \( 1 - 2.43e33T + 1.20e67T^{2} \)
79 \( 1 + 4.92e33iT - 2.06e68T^{2} \)
83 \( 1 + 3.24e34iT - 1.22e69T^{2} \)
89 \( 1 - 9.08e34T + 1.50e70T^{2} \)
97 \( 1 + 1.10e36T + 3.34e71T^{2} \)
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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

−16.45948575211506164857068844214, −13.79978355424930679831326956490, −12.72347022342638791370862836790, −10.50978308704405337276228630339, −9.763275073376664222134909156723, −7.71360685147866186813200502721, −6.41850581088994271442283702099, −3.81677890688655224351583529294, −1.66740501937398034412756259438, −1.19712069706608386776815285995, 1.15150936161162853110955585664, 2.52851057964172917842280157606, 5.42740470668675607452650908615, 6.30373292173909526867533883680, 8.707713639950289705747409159145, 9.544696956549945957588854189111, 11.04580455606564223851134795464, 13.45239696329506058791948450945, 15.21256456951806077891984968293, 16.33010754035220656565919390605

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