↑

Properties

Label 2-2e2-4.3-c38-0-11
Degree $2$
Conductor $4$
Sign $-0.494 + 0.869i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.53e5 + 2.63e5i)2-s − 7.71e8i·3-s + (1.35e11 − 2.38e11i)4-s − 2.10e13·5-s + (2.03e14 + 3.49e14i)6-s − 5.85e15i·7-s + (1.33e15 + 1.44e17i)8-s + 7.56e17·9-s + (9.54e18 − 5.54e18i)10-s − 6.60e19i·11-s + (−1.84e20 − 1.04e20i)12-s + 1.72e21·13-s + (1.54e21 + 2.65e21i)14-s + 1.62e22i·15-s + (−3.85e22 − 6.49e22i)16-s + 2.34e23·17-s + ⋯
L(s)  = 1  + (−0.864 + 0.502i)2-s − 0.663i·3-s + (0.494 − 0.869i)4-s − 1.10·5-s + (0.333 + 0.573i)6-s − 0.513i·7-s + (0.00925 + 0.999i)8-s + 0.559·9-s + (0.954 − 0.554i)10-s − 1.07i·11-s + (−0.576 − 0.328i)12-s + 1.18·13-s + (0.258 + 0.443i)14-s + 0.732i·15-s + (−0.510 − 0.859i)16-s + 0.980·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.494 + 0.869i$
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: \((2,\ 4,\ (\ :19),\ -0.494 + 0.869i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.8939295887\)
\(L(\frac12)\) \(\approx\) \(0.8939295887\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.53e5 - 2.63e5i)T \)
good3 \( 1 + 7.71e8iT - 1.35e18T^{2} \)
5 \( 1 + 2.10e13T + 3.63e26T^{2} \)
7 \( 1 + 5.85e15iT - 1.29e32T^{2} \)
11 \( 1 + 6.60e19iT - 3.74e39T^{2} \)
13 \( 1 - 1.72e21T + 2.13e42T^{2} \)
17 \( 1 - 2.34e23T + 5.71e46T^{2} \)
19 \( 1 - 2.63e24iT - 3.91e48T^{2} \)
23 \( 1 - 9.74e23iT - 5.56e51T^{2} \)
29 \( 1 - 4.55e27T + 3.72e55T^{2} \)
31 \( 1 + 2.63e28iT - 4.69e56T^{2} \)
37 \( 1 + 1.68e29T + 3.90e59T^{2} \)
41 \( 1 + 3.77e30T + 1.93e61T^{2} \)
43 \( 1 + 2.13e31iT - 1.17e62T^{2} \)
47 \( 1 - 1.21e31iT - 3.46e63T^{2} \)
53 \( 1 + 5.52e32T + 3.33e65T^{2} \)
59 \( 1 - 3.38e33iT - 1.96e67T^{2} \)
61 \( 1 - 2.82e33T + 6.95e67T^{2} \)
67 \( 1 + 9.12e34iT - 2.45e69T^{2} \)
71 \( 1 - 3.38e34iT - 2.22e70T^{2} \)
73 \( 1 + 4.50e35T + 6.40e70T^{2} \)
79 \( 1 + 1.73e36iT - 1.28e72T^{2} \)
83 \( 1 - 2.19e36iT - 8.41e72T^{2} \)
89 \( 1 + 1.76e37T + 1.19e74T^{2} \)
97 \( 1 + 1.37e37T + 3.14e75T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.63070049557876262636075899201, −13.84281586696000782792932554986, −11.85164461203358512146370635453, −10.42094261367322709245427604556, −8.349953915991841492535735549971, −7.49863216508466548260759811494, −6.04542821788819248168603205526, −3.71433885793444652158444062275, −1.37887830299007690066378176719, −0.42490164090191662890763646796, 1.27640759456862702241595419147, 3.17920377951517740988129875542, 4.44658628892469262184965576660, 7.10252903142604318386780616428, 8.539757286875393084201601101158, 9.947651225743739418780501090951, 11.33162506381214749682002053331, 12.59194526158141570842808646202, 15.38305036249569745696312509531, 16.08326662041712062470564344228

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