Properties

Label 2-6-3.2-c24-0-6
Degree $2$
Conductor $6$
Sign $0.454 + 0.890i$
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 + (4.73e5 − 2.41e5i)3-s − 8.38e6·4-s − 7.85e7i·5-s + (7.00e8 + 1.37e9i)6-s + 2.19e9·7-s − 2.42e10i·8-s + (1.65e11 − 2.28e11i)9-s + 2.27e11·10-s − 1.55e12i·11-s + (−3.97e12 + 2.02e12i)12-s − 3.72e13·13-s + 6.36e12i·14-s + (−1.89e13 − 3.71e13i)15-s + 7.03e13·16-s − 6.73e14i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.890 − 0.454i)3-s − 0.500·4-s − 0.321i·5-s + (0.321 + 0.629i)6-s + 0.158·7-s − 0.353i·8-s + (0.586 − 0.810i)9-s + 0.227·10-s − 0.496i·11-s + (−0.445 + 0.227i)12-s − 1.59·13-s + 0.112i·14-s + (−0.146 − 0.286i)15-s + 0.250·16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.454 + 0.890i$
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: \((2,\ 6,\ (\ :12),\ 0.454 + 0.890i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(1.72435 - 1.05549i\)
\(L(\frac12)\) \(\approx\) \(1.72435 - 1.05549i\)
\(L(13)\) 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 \( 1 + (-4.73e5 + 2.41e5i)T \)
good5 \( 1 + 7.85e7iT - 5.96e16T^{2} \)
7 \( 1 - 2.19e9T + 1.91e20T^{2} \)
11 \( 1 + 1.55e12iT - 9.84e24T^{2} \)
13 \( 1 + 3.72e13T + 5.42e26T^{2} \)
17 \( 1 + 6.73e14iT - 3.39e29T^{2} \)
19 \( 1 - 9.37e14T + 4.89e30T^{2} \)
23 \( 1 + 3.13e16iT - 4.80e32T^{2} \)
29 \( 1 + 2.21e17iT - 1.25e35T^{2} \)
31 \( 1 - 6.76e17T + 6.20e35T^{2} \)
37 \( 1 + 4.23e18T + 4.33e37T^{2} \)
41 \( 1 - 2.43e19iT - 5.09e38T^{2} \)
43 \( 1 - 2.27e19T + 1.59e39T^{2} \)
47 \( 1 + 6.93e19iT - 1.35e40T^{2} \)
53 \( 1 + 8.43e20iT - 2.41e41T^{2} \)
59 \( 1 - 2.79e21iT - 3.16e42T^{2} \)
61 \( 1 + 4.56e21T + 7.04e42T^{2} \)
67 \( 1 + 1.20e21T + 6.69e43T^{2} \)
71 \( 1 - 1.60e22iT - 2.69e44T^{2} \)
73 \( 1 - 2.60e22T + 5.24e44T^{2} \)
79 \( 1 - 8.00e22T + 3.49e45T^{2} \)
83 \( 1 - 1.65e23iT - 1.14e46T^{2} \)
89 \( 1 - 4.33e23iT - 6.10e46T^{2} \)
97 \( 1 + 4.08e23T + 4.81e47T^{2} \)
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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

−16.52809473288082874578905395470, −14.91252857120113377276869105782, −13.81374831013796575947817601374, −12.30450691327406965339705891210, −9.609288653392221954546668050369, −8.218948577989580171774736738315, −6.89169364143464593024218752702, −4.77507282208977231312617984814, −2.70409848570300123722015936925, −0.63596541731637220328247265360, 1.82589380788275317168779895448, 3.19496260735458942556410499950, 4.77829198380849522749416059564, 7.59617104210935186332149052989, 9.336146025282551463308320041271, 10.53389384276533900403373585603, 12.45561452705865824372249772085, 14.11604115623435619593383722593, 15.24591637721839846821562116699, 17.33948114540955916834647246529

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