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Properties

Label 2-6-3.2-c24-0-4
Degree $2$
Conductor $6$
Sign $0.801 + 0.598i$
Analytic cond. $21.8980$
Root an. cond. $4.67953$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.89e3i·2-s + (−3.17e5 + 4.25e5i)3-s − 8.38e6·4-s − 6.76e7i·5-s + (1.23e9 + 9.20e8i)6-s − 2.41e10·7-s + 2.42e10i·8-s + (−8.02e10 − 2.70e11i)9-s − 1.96e11·10-s + 4.00e12i·11-s + (2.66e12 − 3.57e12i)12-s + 2.67e13·13-s + 6.99e13i·14-s + (2.88e13 + 2.15e13i)15-s + 7.03e13·16-s − 6.60e14i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.598 + 0.801i)3-s − 0.500·4-s − 0.277i·5-s + (0.566 + 0.423i)6-s − 1.74·7-s + 0.353i·8-s + (−0.284 − 0.958i)9-s − 0.196·10-s + 1.27i·11-s + (0.299 − 0.400i)12-s + 1.14·13-s + 1.23i·14-s + (0.222 + 0.165i)15-s + 0.250·16-s − 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.801 + 0.598i$
Analytic conductor: \(21.8980\)
Root analytic conductor: \(4.67953\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :12),\ 0.801 + 0.598i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.942450 - 0.313018i\)
\(L(\frac12)\) \(\approx\) \(0.942450 - 0.313018i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.89e3iT \)
3 \( 1 + (3.17e5 - 4.25e5i)T \)
good5 \( 1 + 6.76e7iT - 5.96e16T^{2} \)
7 \( 1 + 2.41e10T + 1.91e20T^{2} \)
11 \( 1 - 4.00e12iT - 9.84e24T^{2} \)
13 \( 1 - 2.67e13T + 5.42e26T^{2} \)
17 \( 1 + 6.60e14iT - 3.39e29T^{2} \)
19 \( 1 + 3.05e14T + 4.89e30T^{2} \)
23 \( 1 + 7.66e14iT - 4.80e32T^{2} \)
29 \( 1 + 4.00e17iT - 1.25e35T^{2} \)
31 \( 1 - 1.08e18T + 6.20e35T^{2} \)
37 \( 1 - 2.67e18T + 4.33e37T^{2} \)
41 \( 1 - 3.10e19iT - 5.09e38T^{2} \)
43 \( 1 + 2.43e19T + 1.59e39T^{2} \)
47 \( 1 + 3.79e19iT - 1.35e40T^{2} \)
53 \( 1 + 8.82e19iT - 2.41e41T^{2} \)
59 \( 1 - 1.88e21iT - 3.16e42T^{2} \)
61 \( 1 + 1.92e21T + 7.04e42T^{2} \)
67 \( 1 + 7.96e21T + 6.69e43T^{2} \)
71 \( 1 + 2.04e22iT - 2.69e44T^{2} \)
73 \( 1 - 3.28e22T + 5.24e44T^{2} \)
79 \( 1 - 5.13e21T + 3.49e45T^{2} \)
83 \( 1 - 1.14e21iT - 1.14e46T^{2} \)
89 \( 1 + 3.69e23iT - 6.10e46T^{2} \)
97 \( 1 - 1.10e24T + 4.81e47T^{2} \)
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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

−16.65174454861606567790148718490, −15.49241698686138553085991824011, −13.20838237916596027399257910963, −11.91575273344811772176171136136, −10.18636110977645162462806914311, −9.280003670636718388586550580495, −6.37252564123181990505634955493, −4.51683358311718038156666834832, −3.06021780283389970378503335732, −0.61567017855446826661169655473, 0.76899749529298679323563685936, 3.32089522526169056143070494622, 5.94896674554765800587462846112, 6.64438169167547012397760112823, 8.562799986798889794259426651545, 10.67251086298437860719355757207, 12.70830870422816203963260997921, 13.71767763370649105117867881516, 15.90156874703210938614560529320, 16.84379395263954427402307844880

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