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Properties

Label 2-6-3.2-c20-0-5
Degree $2$
Conductor $6$
Sign $-0.706 - 0.707i$
Analytic cond. $15.2108$
Root an. cond. $3.90010$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 724. i·2-s + (4.17e4 − 4.17e4i)3-s − 5.24e5·4-s + 8.05e5i·5-s + (−3.02e7 − 3.02e7i)6-s − 3.63e8·7-s + 3.79e8i·8-s + (6.43e6 − 3.48e9i)9-s + 5.83e8·10-s − 7.52e9i·11-s + (−2.19e10 + 2.18e10i)12-s − 1.09e11·13-s + 2.63e11i·14-s + (3.36e10 + 3.36e10i)15-s + 2.74e11·16-s + 2.97e12i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.707 − 0.706i)3-s − 0.500·4-s + 0.0824i·5-s + (−0.499 − 0.500i)6-s − 1.28·7-s + 0.353i·8-s + (0.00184 − 0.999i)9-s + 0.0583·10-s − 0.290i·11-s + (−0.353 + 0.353i)12-s − 0.794·13-s + 0.910i·14-s + (0.0582 + 0.0583i)15-s + 0.250·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.706 - 0.707i$
Analytic conductor: \(15.2108\)
Root analytic conductor: \(3.90010\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :10),\ -0.706 - 0.707i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.236029 + 0.569083i\)
\(L(\frac12)\) \(\approx\) \(0.236029 + 0.569083i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 724. iT \)
3 \( 1 + (-4.17e4 + 4.17e4i)T \)
good5 \( 1 - 8.05e5iT - 9.53e13T^{2} \)
7 \( 1 + 3.63e8T + 7.97e16T^{2} \)
11 \( 1 + 7.52e9iT - 6.72e20T^{2} \)
13 \( 1 + 1.09e11T + 1.90e22T^{2} \)
17 \( 1 - 2.97e12iT - 4.06e24T^{2} \)
19 \( 1 + 9.19e12T + 3.75e25T^{2} \)
23 \( 1 - 8.43e12iT - 1.71e27T^{2} \)
29 \( 1 + 6.14e14iT - 1.76e29T^{2} \)
31 \( 1 - 5.36e14T + 6.71e29T^{2} \)
37 \( 1 + 6.40e15T + 2.31e31T^{2} \)
41 \( 1 + 1.51e16iT - 1.80e32T^{2} \)
43 \( 1 + 2.28e16T + 4.67e32T^{2} \)
47 \( 1 + 2.22e16iT - 2.76e33T^{2} \)
53 \( 1 + 3.38e17iT - 3.05e34T^{2} \)
59 \( 1 - 3.93e17iT - 2.61e35T^{2} \)
61 \( 1 - 7.29e17T + 5.08e35T^{2} \)
67 \( 1 + 5.04e17T + 3.32e36T^{2} \)
71 \( 1 + 6.11e18iT - 1.05e37T^{2} \)
73 \( 1 - 9.17e17T + 1.84e37T^{2} \)
79 \( 1 + 7.30e18T + 8.96e37T^{2} \)
83 \( 1 - 1.57e19iT - 2.40e38T^{2} \)
89 \( 1 - 2.49e19iT - 9.72e38T^{2} \)
97 \( 1 - 8.81e19T + 5.43e39T^{2} \)
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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

−17.20673619994875009447252963105, −14.98699417522882829409085383732, −13.31229560012509081894096398944, −12.35833878199831770686943317100, −10.13177914606164511697406719493, −8.552479299683577747152034448558, −6.51216389938073908373882127062, −3.61671645940159064098465790208, −2.17813237426377690151933394334, −0.21256879817809959139116301500, 2.95081155686897978026254270354, 4.76811473455732518614688099133, 6.93949357812049839525907219290, 8.876456029836846094365247340503, 10.09262225756243317754488616636, 12.86297578833362028357146033446, 14.41351311421614390923984060622, 15.74479207371612493245027953600, 16.79264487026728351241448238238, 18.90560854340368257170756294920

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