Properties

Label 2-6-3.2-c30-0-5
Degree $2$
Conductor $6$
Sign $0.901 - 0.433i$
Analytic cond. $34.2085$
Root an. cond. $5.84880$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.31e4i·2-s + (6.21e6 + 1.29e7i)3-s − 5.36e8·4-s − 2.12e10i·5-s + (−2.99e11 + 1.43e11i)6-s − 1.34e12·7-s − 1.24e13i·8-s + (−1.28e14 + 1.60e14i)9-s + 4.92e14·10-s − 4.92e15i·11-s + (−3.33e15 − 6.94e15i)12-s + 6.88e16·13-s − 3.11e16i·14-s + (2.75e17 − 1.32e17i)15-s + 2.88e17·16-s − 1.65e18i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.433 + 0.901i)3-s − 0.500·4-s − 0.697i·5-s + (−0.637 + 0.306i)6-s − 0.283·7-s − 0.353i·8-s + (−0.624 + 0.780i)9-s + 0.492·10-s − 1.17i·11-s + (−0.216 − 0.450i)12-s + 1.34·13-s − 0.200i·14-s + (0.628 − 0.301i)15-s + 0.250·16-s − 0.576i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.901 - 0.433i$
Analytic conductor: \(34.2085\)
Root analytic conductor: \(5.84880\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :15),\ 0.901 - 0.433i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(2.050338628\)
\(L(\frac12)\) \(\approx\) \(2.050338628\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.31e4iT \)
3 \( 1 + (-6.21e6 - 1.29e7i)T \)
good5 \( 1 + 2.12e10iT - 9.31e20T^{2} \)
7 \( 1 + 1.34e12T + 2.25e25T^{2} \)
11 \( 1 + 4.92e15iT - 1.74e31T^{2} \)
13 \( 1 - 6.88e16T + 2.61e33T^{2} \)
17 \( 1 + 1.65e18iT - 8.19e36T^{2} \)
19 \( 1 - 1.39e17T + 2.30e38T^{2} \)
23 \( 1 + 3.70e20iT - 7.10e40T^{2} \)
29 \( 1 - 9.68e21iT - 7.44e43T^{2} \)
31 \( 1 + 8.03e20T + 5.50e44T^{2} \)
37 \( 1 - 5.84e23T + 1.11e47T^{2} \)
41 \( 1 - 1.21e24iT - 2.41e48T^{2} \)
43 \( 1 - 4.57e24T + 1.00e49T^{2} \)
47 \( 1 + 4.58e24iT - 1.45e50T^{2} \)
53 \( 1 + 5.51e25iT - 5.34e51T^{2} \)
59 \( 1 - 3.03e25iT - 1.33e53T^{2} \)
61 \( 1 + 9.38e26T + 3.62e53T^{2} \)
67 \( 1 - 2.46e27T + 6.05e54T^{2} \)
71 \( 1 + 4.09e26iT - 3.44e55T^{2} \)
73 \( 1 - 7.27e24T + 7.93e55T^{2} \)
79 \( 1 - 4.60e28T + 8.48e56T^{2} \)
83 \( 1 + 1.19e29iT - 3.73e57T^{2} \)
89 \( 1 + 2.72e29iT - 3.03e58T^{2} \)
97 \( 1 - 5.73e29T + 4.01e59T^{2} \)
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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

−16.09552587257867705565338207487, −14.42636525072897021480967075578, −13.17239127732090213962456655671, −10.90813903810490614708442385876, −9.142677542222589868825206711271, −8.298849728633956596964783359554, −6.05711062753439968401274381007, −4.63167493470376248418122327044, −3.21343794988076543233983882944, −0.69974980213164739090286033849, 1.15989718197870355791219246871, 2.42091599862527068717879399475, 3.76370482499257785293209984783, 6.21520075948494077575798988858, 7.73162115576151935309969354557, 9.417386276521494679933274843588, 11.10215209706040161560853071391, 12.58124676105593836140123707357, 13.73085824387247660814193179361, 15.16298019422291938090781309591

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