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Properties

Label 2-2e2-4.3-c38-0-15
Degree $2$
Conductor $4$
Sign $-0.748 + 0.662i$
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  + (4.90e5 − 1.85e5i)2-s + 6.63e7i·3-s + (2.05e11 − 1.82e11i)4-s − 1.96e13·5-s + (1.23e13 + 3.25e13i)6-s + 1.51e16i·7-s + (6.70e16 − 1.27e17i)8-s + 1.34e18·9-s + (−9.63e18 + 3.65e18i)10-s − 1.12e20i·11-s + (1.20e19 + 1.36e19i)12-s − 5.44e20·13-s + (2.81e21 + 7.42e21i)14-s − 1.30e21i·15-s + (9.20e21 − 7.49e22i)16-s − 3.53e23·17-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)2-s + 0.0570i·3-s + (0.748 − 0.662i)4-s − 1.03·5-s + (0.0202 + 0.0533i)6-s + 1.32i·7-s + (0.465 − 0.885i)8-s + 0.996·9-s + (−0.963 + 0.365i)10-s − 1.83i·11-s + (0.0378 + 0.0427i)12-s − 0.372·13-s + (0.470 + 1.24i)14-s − 0.0587i·15-s + (0.121 − 0.992i)16-s − 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.748 + 0.662i$
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: \((2,\ 4,\ (\ :19),\ -0.748 + 0.662i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(1.981004082\)
\(L(\frac12)\) \(\approx\) \(1.981004082\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.90e5 + 1.85e5i)T \)
good3 \( 1 - 6.63e7iT - 1.35e18T^{2} \)
5 \( 1 + 1.96e13T + 3.63e26T^{2} \)
7 \( 1 - 1.51e16iT - 1.29e32T^{2} \)
11 \( 1 + 1.12e20iT - 3.74e39T^{2} \)
13 \( 1 + 5.44e20T + 2.13e42T^{2} \)
17 \( 1 + 3.53e23T + 5.71e46T^{2} \)
19 \( 1 + 2.66e24iT - 3.91e48T^{2} \)
23 \( 1 + 2.65e25iT - 5.56e51T^{2} \)
29 \( 1 + 1.06e27T + 3.72e55T^{2} \)
31 \( 1 - 6.94e27iT - 4.69e56T^{2} \)
37 \( 1 + 9.31e29T + 3.90e59T^{2} \)
41 \( 1 + 4.70e29T + 1.93e61T^{2} \)
43 \( 1 + 8.70e30iT - 1.17e62T^{2} \)
47 \( 1 + 5.03e31iT - 3.46e63T^{2} \)
53 \( 1 - 8.75e32T + 3.33e65T^{2} \)
59 \( 1 + 5.29e33iT - 1.96e67T^{2} \)
61 \( 1 - 4.70e33T + 6.95e67T^{2} \)
67 \( 1 + 3.40e34iT - 2.45e69T^{2} \)
71 \( 1 + 1.85e34iT - 2.22e70T^{2} \)
73 \( 1 + 1.08e35T + 6.40e70T^{2} \)
79 \( 1 - 1.22e36iT - 1.28e72T^{2} \)
83 \( 1 - 3.51e36iT - 8.41e72T^{2} \)
89 \( 1 - 1.15e37T + 1.19e74T^{2} \)
97 \( 1 - 1.01e37T + 3.14e75T^{2} \)
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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

−15.33284340764665415568169464731, −13.41968878053976495240189491690, −12.01565359909743353708419469501, −10.98247169001492742697347672014, −8.761400710806164342002750147004, −6.75638365116681754778588920605, −5.14270624818975871334135980644, −3.71273580145585222034100337492, −2.34871790383664674553175472770, −0.40065507922796943491027907760, 1.77576748748156491110950592502, 3.96266315554836076624948889993, 4.48925193170877236419886950729, 6.99931794603655025095124711729, 7.58084973585226728178315952954, 10.33118299707035431537580894184, 12.03076562026504339198234394080, 13.20421842905967320806484832478, 14.88241780780808112646496047476, 15.94944090441452364005736708053

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