Properties

Degree 2
Conductor $ 2^{2} $
Sign $0.104 - 0.994i$
Motivic weight 36
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94e5 − 1.75e5i)2-s − 4.34e8i·3-s + (7.18e9 − 6.83e10i)4-s − 7.52e12·5-s + (−7.62e13 − 8.47e13i)6-s − 2.26e15i·7-s + (−1.05e16 − 1.45e16i)8-s − 3.89e16·9-s + (−1.46e18 + 1.31e18i)10-s − 3.43e17i·11-s + (−2.97e19 − 3.12e18i)12-s + 4.11e19·13-s + (−3.97e20 − 4.41e20i)14-s + 3.27e21i·15-s + (−4.61e21 − 9.82e20i)16-s + 1.36e22·17-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s − 1.12i·3-s + (0.104 − 0.994i)4-s − 1.97·5-s + (−0.751 − 0.834i)6-s − 1.39i·7-s + (−0.587 − 0.809i)8-s − 0.259·9-s + (−1.46 + 1.31i)10-s − 0.0617i·11-s + (−1.11 − 0.117i)12-s + 0.365·13-s + (−0.930 − 1.03i)14-s + 2.21i·15-s + (−0.978 − 0.208i)16-s + 0.972·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $0.104 - 0.994i$
motivic weight  =  \(36\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :18),\ 0.104 - 0.994i)\)
\(L(\frac{37}{2})\)  \(\approx\)  \(1.321501159\)
\(L(\frac12)\)  \(\approx\)  \(1.321501159\)
\(L(19)\)   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 \( 1 + (-1.94e5 + 1.75e5i)T \)
good3 \( 1 + 4.34e8iT - 1.50e17T^{2} \)
5 \( 1 + 7.52e12T + 1.45e25T^{2} \)
7 \( 1 + 2.26e15iT - 2.65e30T^{2} \)
11 \( 1 + 3.43e17iT - 3.09e37T^{2} \)
13 \( 1 - 4.11e19T + 1.26e40T^{2} \)
17 \( 1 - 1.36e22T + 1.97e44T^{2} \)
19 \( 1 - 6.68e21iT - 1.08e46T^{2} \)
23 \( 1 + 3.00e24iT - 1.05e49T^{2} \)
29 \( 1 + 1.64e26T + 4.42e52T^{2} \)
31 \( 1 - 4.32e26iT - 4.88e53T^{2} \)
37 \( 1 + 2.47e27T + 2.85e56T^{2} \)
41 \( 1 - 3.98e28T + 1.14e58T^{2} \)
43 \( 1 - 3.42e29iT - 6.38e58T^{2} \)
47 \( 1 + 7.98e29iT - 1.56e60T^{2} \)
53 \( 1 + 1.60e30T + 1.18e62T^{2} \)
59 \( 1 + 8.75e31iT - 5.63e63T^{2} \)
61 \( 1 + 3.85e31T + 1.87e64T^{2} \)
67 \( 1 - 7.88e32iT - 5.47e65T^{2} \)
71 \( 1 + 3.14e33iT - 4.41e66T^{2} \)
73 \( 1 + 3.43e33T + 1.20e67T^{2} \)
79 \( 1 - 9.73e33iT - 2.06e68T^{2} \)
83 \( 1 + 3.05e34iT - 1.22e69T^{2} \)
89 \( 1 - 1.54e35T + 1.50e70T^{2} \)
97 \( 1 + 2.22e35T + 3.34e71T^{2} \)
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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

−14.50667635697009270839213499528, −13.00066681753523727268404649836, −11.94368392108770701218222500317, −10.71870533262132719444865140092, −7.85730878484204942387844550259, −6.83837428417473833291642958683, −4.37859950966259184475433839311, −3.35417896003519484497123416613, −1.18036922855491536329562800152, −0.36659457287038227066141878196, 3.20729343278486458351547952298, 4.12574245572111724383286374452, 5.42061816630855606782404890278, 7.57338638196792409140670048807, 8.876792391505694212027050236118, 11.36706225754926965615087341097, 12.37292640099375414632340430704, 14.96009602644291115044848295645, 15.50382533338031385943304480770, 16.36781275879018641886417701543

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