Properties

Label 2-2e2-4.3-c38-0-1
Degree $2$
Conductor $4$
Sign $0.318 + 0.947i$
Analytic cond. $36.5853$
Root an. cond. $6.04858$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.06e5 + 4.25e5i)2-s + 1.26e9i·3-s + (−8.75e10 − 2.60e11i)4-s − 4.90e12·5-s + (−5.38e14 − 3.86e14i)6-s + 1.75e16i·7-s + (1.37e17 + 4.24e16i)8-s − 2.47e17·9-s + (1.50e18 − 2.08e18i)10-s + 5.84e19i·11-s + (3.29e20 − 1.10e20i)12-s − 2.66e21·13-s + (−7.46e21 − 5.36e21i)14-s − 6.19e21i·15-s + (−6.02e22 + 4.56e22i)16-s + 3.72e22·17-s + ⋯
L(s)  = 1  + (−0.583 + 0.811i)2-s + 1.08i·3-s + (−0.318 − 0.947i)4-s − 0.257·5-s + (−0.883 − 0.635i)6-s + 1.53i·7-s + (0.955 + 0.294i)8-s − 0.183·9-s + (0.150 − 0.208i)10-s + 0.956i·11-s + (1.03 − 0.346i)12-s − 1.82·13-s + (−1.24 − 0.897i)14-s − 0.279i·15-s + (−0.797 + 0.603i)16-s + 0.155·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(36.5853\)
Root analytic conductor: \(6.04858\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :19),\ 0.318 + 0.947i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.6040925297\)
\(L(\frac12)\) \(\approx\) \(0.6040925297\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.06e5 - 4.25e5i)T \)
good3 \( 1 - 1.26e9iT - 1.35e18T^{2} \)
5 \( 1 + 4.90e12T + 3.63e26T^{2} \)
7 \( 1 - 1.75e16iT - 1.29e32T^{2} \)
11 \( 1 - 5.84e19iT - 3.74e39T^{2} \)
13 \( 1 + 2.66e21T + 2.13e42T^{2} \)
17 \( 1 - 3.72e22T + 5.71e46T^{2} \)
19 \( 1 - 1.66e24iT - 3.91e48T^{2} \)
23 \( 1 - 1.32e25iT - 5.56e51T^{2} \)
29 \( 1 - 4.91e26T + 3.72e55T^{2} \)
31 \( 1 + 3.21e27iT - 4.69e56T^{2} \)
37 \( 1 - 9.26e29T + 3.90e59T^{2} \)
41 \( 1 - 2.76e30T + 1.93e61T^{2} \)
43 \( 1 + 8.54e30iT - 1.17e62T^{2} \)
47 \( 1 + 1.63e31iT - 3.46e63T^{2} \)
53 \( 1 - 1.08e33T + 3.33e65T^{2} \)
59 \( 1 - 6.34e33iT - 1.96e67T^{2} \)
61 \( 1 + 9.47e33T + 6.95e67T^{2} \)
67 \( 1 + 5.59e34iT - 2.45e69T^{2} \)
71 \( 1 + 2.26e35iT - 2.22e70T^{2} \)
73 \( 1 - 7.29e34T + 6.40e70T^{2} \)
79 \( 1 - 7.48e35iT - 1.28e72T^{2} \)
83 \( 1 + 3.83e36iT - 8.41e72T^{2} \)
89 \( 1 + 1.37e37T + 1.19e74T^{2} \)
97 \( 1 - 2.37e37T + 3.14e75T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.70488847427207662816334108207, −15.32815048936868591075778685838, −14.85708538300682434473408142825, −12.13425017785085356371053477717, −10.04643690231727583158812530089, −9.248569257183170044227989260823, −7.56662063690893552145590308552, −5.60347486693340197415821253646, −4.49585605044733993901973457020, −2.17999902882479998201321659116, 0.25766913167989487359915413919, 1.02247560921973326275196200675, 2.58726603084230737902500152695, 4.25960465558486300318781738767, 7.07306127055173823598953531018, 7.86825752886076646870351302693, 9.900439744874242301138238681978, 11.38021080114459753829523434424, 12.78998395933352432732902400414, 13.88299168442832185204757098693

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