Properties

Degree 2
Conductor $ 2^{2} $
Sign $0.985 + 0.168i$
Motivic weight 36
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.61e5 + 2.22e4i)2-s + 2.96e8i·3-s + (6.77e10 + 1.16e10i)4-s + 1.46e12·5-s + (−6.58e12 + 7.73e13i)6-s − 2.38e15i·7-s + (1.74e16 + 4.53e15i)8-s + 6.23e16·9-s + (3.82e17 + 3.25e16i)10-s − 1.01e19i·11-s + (−3.43e18 + 2.00e19i)12-s − 4.81e19·13-s + (5.30e19 − 6.23e20i)14-s + 4.33e20i·15-s + (4.45e21 + 1.57e21i)16-s + 1.19e22·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0847i)2-s + 0.764i·3-s + (0.985 + 0.168i)4-s + 0.383·5-s + (−0.0648 + 0.761i)6-s − 1.46i·7-s + (0.967 + 0.251i)8-s + 0.415·9-s + (0.382 + 0.0325i)10-s − 1.81i·11-s + (−0.129 + 0.753i)12-s − 0.427·13-s + (0.124 − 1.46i)14-s + 0.293i·15-s + (0.942 + 0.333i)16-s + 0.850·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $0.985 + 0.168i$
motivic weight  =  \(36\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :18),\ 0.985 + 0.168i)\)
\(L(\frac{37}{2})\)  \(\approx\)  \(4.489267042\)
\(L(\frac12)\)  \(\approx\)  \(4.489267042\)
\(L(19)\)   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 \( 1 + (-2.61e5 - 2.22e4i)T \)
good3 \( 1 - 2.96e8iT - 1.50e17T^{2} \)
5 \( 1 - 1.46e12T + 1.45e25T^{2} \)
7 \( 1 + 2.38e15iT - 2.65e30T^{2} \)
11 \( 1 + 1.01e19iT - 3.09e37T^{2} \)
13 \( 1 + 4.81e19T + 1.26e40T^{2} \)
17 \( 1 - 1.19e22T + 1.97e44T^{2} \)
19 \( 1 + 1.66e22iT - 1.08e46T^{2} \)
23 \( 1 - 2.63e24iT - 1.05e49T^{2} \)
29 \( 1 - 1.36e26T + 4.42e52T^{2} \)
31 \( 1 + 4.52e26iT - 4.88e53T^{2} \)
37 \( 1 - 5.31e27T + 2.85e56T^{2} \)
41 \( 1 - 1.73e29T + 1.14e58T^{2} \)
43 \( 1 - 2.26e29iT - 6.38e58T^{2} \)
47 \( 1 + 1.08e30iT - 1.56e60T^{2} \)
53 \( 1 + 1.23e31T + 1.18e62T^{2} \)
59 \( 1 - 9.54e31iT - 5.63e63T^{2} \)
61 \( 1 + 5.38e31T + 1.87e64T^{2} \)
67 \( 1 + 6.26e32iT - 5.47e65T^{2} \)
71 \( 1 - 2.83e33iT - 4.41e66T^{2} \)
73 \( 1 + 6.34e33T + 1.20e67T^{2} \)
79 \( 1 - 2.21e34iT - 2.06e68T^{2} \)
83 \( 1 + 6.15e34iT - 1.22e69T^{2} \)
89 \( 1 + 8.78e34T + 1.50e70T^{2} \)
97 \( 1 - 1.81e35T + 3.34e71T^{2} \)
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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

−16.00215723937804429577959467180, −14.25990980299999720603577761408, −13.30028666865640048720277312798, −11.19965280015531061816180123152, −10.04104841813583769191198408959, −7.55417062322113787388420231550, −5.84629944555684351116745461542, −4.30273132739199656197371463377, −3.25973076986313208690850435046, −1.06896055379920513725414306427, 1.66782833515984445829227209120, 2.49806010945054053509721309488, 4.70394974500679176404194006159, 6.12582981616502223385106950463, 7.50177532050709937022560789708, 9.869518858813929121769505578823, 12.19197052102525829641922877602, 12.62451949054900662122080757705, 14.46200044080201246490929633427, 15.65780241003025340486809186645

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