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Properties

Label 2-3-3.2-c64-0-18
Degree $2$
Conductor $3$
Sign $-0.536 - 0.844i$
Analytic cond. $77.8210$
Root an. cond. $8.82162$
Motivic weight $64$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.45e9i·2-s + (−9.93e14 − 1.56e15i)3-s − 1.35e18·4-s − 4.47e22i·5-s + (6.96e24 − 4.42e24i)6-s − 9.42e26·7-s + 7.60e28i·8-s + (−1.45e30 + 3.10e30i)9-s + 1.99e32·10-s − 1.15e33i·11-s + (1.34e33 + 2.12e33i)12-s − 4.26e35·13-s − 4.19e36i·14-s + (−6.99e37 + 4.44e37i)15-s − 3.63e38·16-s − 4.50e39i·17-s + ⋯
L(s)  = 1  + 1.03i·2-s + (−0.536 − 0.844i)3-s − 0.0736·4-s − 1.92i·5-s + (0.874 − 0.555i)6-s − 0.853·7-s + 0.959i·8-s + (−0.424 + 0.905i)9-s + 1.99·10-s − 0.545i·11-s + (0.0394 + 0.0621i)12-s − 0.963·13-s − 0.884i·14-s + (−1.62 + 1.03i)15-s − 1.06·16-s − 1.90i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.536 - 0.844i$
Analytic conductor: \(77.8210\)
Root analytic conductor: \(8.82162\)
Motivic weight: \(64\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :32),\ -0.536 - 0.844i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(0.05165841030\)
\(L(\frac12)\) \(\approx\) \(0.05165841030\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (9.93e14 + 1.56e15i)T \)
good2 \( 1 - 4.45e9iT - 1.84e19T^{2} \)
5 \( 1 + 4.47e22iT - 5.42e44T^{2} \)
7 \( 1 + 9.42e26T + 1.21e54T^{2} \)
11 \( 1 + 1.15e33iT - 4.45e66T^{2} \)
13 \( 1 + 4.26e35T + 1.96e71T^{2} \)
17 \( 1 + 4.50e39iT - 5.60e78T^{2} \)
19 \( 1 + 8.89e39T + 6.92e81T^{2} \)
23 \( 1 + 1.53e43iT - 1.41e87T^{2} \)
29 \( 1 + 5.61e45iT - 3.92e93T^{2} \)
31 \( 1 + 1.62e47T + 2.79e95T^{2} \)
37 \( 1 + 4.66e49T + 2.31e100T^{2} \)
41 \( 1 + 1.59e51iT - 1.65e103T^{2} \)
43 \( 1 - 1.19e52T + 3.48e104T^{2} \)
47 \( 1 - 3.71e53iT - 1.03e107T^{2} \)
53 \( 1 + 2.17e55iT - 2.25e110T^{2} \)
59 \( 1 - 6.31e56iT - 2.16e113T^{2} \)
61 \( 1 - 1.65e57T + 1.82e114T^{2} \)
67 \( 1 - 3.39e58T + 7.39e116T^{2} \)
71 \( 1 + 1.19e59iT - 3.02e118T^{2} \)
73 \( 1 + 9.39e58T + 1.78e119T^{2} \)
79 \( 1 - 1.16e59T + 2.80e121T^{2} \)
83 \( 1 - 2.45e61iT - 6.62e122T^{2} \)
89 \( 1 - 3.73e62iT - 5.76e124T^{2} \)
97 \( 1 - 6.84e62T + 1.42e127T^{2} \)
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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

−12.68858186221702395923454210357, −11.68104379878570161007675669505, −9.296099091085803715325001253448, −8.063138645211992373668641870306, −6.92240972948747496250693996191, −5.60910459181193290494351980873, −4.87431186191433627955736695684, −2.37691510412411864155552634839, −0.812723871404030884256086339850, −0.01611247201443771454824453849, 2.07569539679621202240763890395, 3.16458792579882869846797897278, 3.91878545594507896421789900173, 6.11779896639080587365051505999, 7.03650303321611709184553548768, 9.818545862244777003954860457133, 10.33789824975690503201737208577, 11.29100024295932964951790929090, 12.56770375919149980897952524976, 14.69339817844787930136791243663

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