Properties

Label 2-3-3.2-c34-0-3
Degree $2$
Conductor $3$
Sign $-0.863 - 0.503i$
Analytic cond. $21.9676$
Root an. cond. $4.68697$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.60e4i·2-s + (1.11e8 + 6.50e7i)3-s + 1.50e10·4-s + 1.37e12i·5-s + (−2.99e12 + 5.14e12i)6-s − 2.89e14·7-s + 1.48e15i·8-s + (8.21e15 + 1.45e16i)9-s − 6.33e16·10-s − 2.11e17i·11-s + (1.67e18 + 9.79e17i)12-s + 1.85e18·13-s − 1.33e19i·14-s + (−8.94e19 + 1.53e20i)15-s + 1.90e20·16-s − 1.25e21i·17-s + ⋯
L(s)  = 1  + 0.351i·2-s + (0.863 + 0.503i)3-s + 0.876·4-s + 1.80i·5-s + (−0.177 + 0.303i)6-s − 1.24·7-s + 0.659i·8-s + (0.492 + 0.870i)9-s − 0.633·10-s − 0.419i·11-s + (0.757 + 0.441i)12-s + 0.214·13-s − 0.437i·14-s + (−0.907 + 1.55i)15-s + 0.644·16-s − 1.51i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.863 - 0.503i$
Analytic conductor: \(21.9676\)
Root analytic conductor: \(4.68697\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :17),\ -0.863 - 0.503i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.684374 + 2.53313i\)
\(L(\frac12)\) \(\approx\) \(0.684374 + 2.53313i\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11e8 - 6.50e7i)T \)
good2 \( 1 - 4.60e4iT - 1.71e10T^{2} \)
5 \( 1 - 1.37e12iT - 5.82e23T^{2} \)
7 \( 1 + 2.89e14T + 5.41e28T^{2} \)
11 \( 1 + 2.11e17iT - 2.55e35T^{2} \)
13 \( 1 - 1.85e18T + 7.48e37T^{2} \)
17 \( 1 + 1.25e21iT - 6.84e41T^{2} \)
19 \( 1 + 1.89e21T + 3.00e43T^{2} \)
23 \( 1 - 9.09e22iT - 1.98e46T^{2} \)
29 \( 1 - 1.52e24iT - 5.26e49T^{2} \)
31 \( 1 - 1.41e25T + 5.08e50T^{2} \)
37 \( 1 - 5.80e26T + 2.08e53T^{2} \)
41 \( 1 - 2.01e27iT - 6.83e54T^{2} \)
43 \( 1 + 1.60e27T + 3.45e55T^{2} \)
47 \( 1 - 1.09e28iT - 7.10e56T^{2} \)
53 \( 1 - 2.12e29iT - 4.22e58T^{2} \)
59 \( 1 - 6.32e29iT - 1.61e60T^{2} \)
61 \( 1 + 1.22e30T + 5.02e60T^{2} \)
67 \( 1 - 1.32e31T + 1.22e62T^{2} \)
71 \( 1 + 6.14e30iT - 8.76e62T^{2} \)
73 \( 1 - 7.66e31T + 2.25e63T^{2} \)
79 \( 1 - 2.85e31T + 3.30e64T^{2} \)
83 \( 1 + 6.49e32iT - 1.77e65T^{2} \)
89 \( 1 + 2.50e31iT - 1.90e66T^{2} \)
97 \( 1 + 4.22e33T + 3.55e67T^{2} \)
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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

−18.71379734497952791094706402833, −16.15798888583157536754939275600, −15.15869291881446128711591201419, −13.85299019920031901576453326188, −11.13644030435663574329210203365, −9.813729598069520156642724164904, −7.46775275388204747361565713918, −6.33819376095564908691154035327, −3.25570128695378748608726378262, −2.60848724103867106882522276323, 0.792540430200967351429202610842, 2.06914229054866310499311459358, 3.83505468001683626233798409857, 6.36242276199249981632185187654, 8.277250272548979369290635164607, 9.741182616260750005503058069342, 12.42767645106724395835285399876, 12.98676472312438076945504570054, 15.48757144607969221078935651563, 16.77511209085016823080842246791

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