Properties

Degree 2
Conductor $ 2^{2} $
Sign $-0.954 - 0.298i$
Motivic weight 36
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.95e4 + 2.59e5i)2-s − 1.35e8i·3-s + (−6.55e10 − 2.04e10i)4-s − 1.24e12·5-s + (3.51e13 + 5.36e12i)6-s + 4.20e14i·7-s + (7.90e15 − 1.61e16i)8-s + 1.31e17·9-s + (4.90e16 − 3.21e17i)10-s − 3.22e18i·11-s + (−2.77e18 + 8.89e18i)12-s − 2.28e19·13-s + (−1.08e20 − 1.66e19i)14-s + 1.68e20i·15-s + (3.88e21 + 2.68e21i)16-s + 6.53e21·17-s + ⋯
L(s)  = 1  + (−0.150 + 0.988i)2-s − 0.350i·3-s + (−0.954 − 0.298i)4-s − 0.325·5-s + (0.346 + 0.0527i)6-s + 0.258i·7-s + (0.438 − 0.898i)8-s + 0.877·9-s + (0.0490 − 0.321i)10-s − 0.579i·11-s + (−0.104 + 0.334i)12-s − 0.203·13-s + (−0.255 − 0.0389i)14-s + 0.113i·15-s + (0.822 + 0.569i)16-s + 0.464·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-0.954 - 0.298i$
motivic weight  =  \(36\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :18),\ -0.954 - 0.298i)\)
\(L(\frac{37}{2})\)  \(\approx\)  \(0.8485647273\)
\(L(\frac12)\)  \(\approx\)  \(0.8485647273\)
\(L(19)\)   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 \( 1 + (3.95e4 - 2.59e5i)T \)
good3 \( 1 + 1.35e8iT - 1.50e17T^{2} \)
5 \( 1 + 1.24e12T + 1.45e25T^{2} \)
7 \( 1 - 4.20e14iT - 2.65e30T^{2} \)
11 \( 1 + 3.22e18iT - 3.09e37T^{2} \)
13 \( 1 + 2.28e19T + 1.26e40T^{2} \)
17 \( 1 - 6.53e21T + 1.97e44T^{2} \)
19 \( 1 - 1.34e23iT - 1.08e46T^{2} \)
23 \( 1 + 4.96e22iT - 1.05e49T^{2} \)
29 \( 1 + 1.19e26T + 4.42e52T^{2} \)
31 \( 1 - 7.37e26iT - 4.88e53T^{2} \)
37 \( 1 + 2.48e28T + 2.85e56T^{2} \)
41 \( 1 + 1.04e29T + 1.14e58T^{2} \)
43 \( 1 - 2.34e29iT - 6.38e58T^{2} \)
47 \( 1 + 6.54e29iT - 1.56e60T^{2} \)
53 \( 1 - 8.42e30T + 1.18e62T^{2} \)
59 \( 1 - 6.65e31iT - 5.63e63T^{2} \)
61 \( 1 + 1.74e32T + 1.87e64T^{2} \)
67 \( 1 - 1.02e33iT - 5.47e65T^{2} \)
71 \( 1 - 3.10e33iT - 4.41e66T^{2} \)
73 \( 1 - 3.37e33T + 1.20e67T^{2} \)
79 \( 1 - 1.58e34iT - 2.06e68T^{2} \)
83 \( 1 - 6.39e34iT - 1.22e69T^{2} \)
89 \( 1 + 1.63e35T + 1.50e70T^{2} \)
97 \( 1 - 3.39e35T + 3.34e71T^{2} \)
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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

−16.61671313333665507998896060816, −15.39407515551796270418868289241, −13.88239859469266406811477417351, −12.35720465589958884882623186301, −10.05354839412489626708119923530, −8.341416828020293844337545739779, −7.09561509100021190180348718714, −5.57656958739945949398556852918, −3.84518782284908815046630527647, −1.33491121188158257291189230602, 0.29639372819466899118334388165, 1.89237082299382099286779909451, 3.64446384261646038410972452633, 4.82731025156185558661027010665, 7.47310374492983621997322779449, 9.333295384455297205278226166733, 10.53043374575063560442399056725, 12.04985125254867634496149951364, 13.46210271741980523109952164476, 15.33879042723034974970817073931

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