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Properties

Label 2-2e2-4.3-c20-0-4
Degree $2$
Conductor $4$
Sign $0.760 - 0.649i$
Analytic cond. $10.1405$
Root an. cond. $3.18442$
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  + (960. − 354. i)2-s + 5.54e4i·3-s + (7.97e5 − 6.81e5i)4-s + 2.53e6·5-s + (1.96e7 + 5.32e7i)6-s + 4.45e8i·7-s + (5.24e8 − 9.37e8i)8-s + 4.15e8·9-s + (2.43e9 − 8.98e8i)10-s + 1.75e10i·11-s + (3.77e10 + 4.41e10i)12-s + 1.80e11·13-s + (1.58e11 + 4.28e11i)14-s + 1.40e11i·15-s + (1.71e11 − 1.08e12i)16-s − 2.03e12·17-s + ⋯
L(s)  = 1  + (0.938 − 0.346i)2-s + 0.938i·3-s + (0.760 − 0.649i)4-s + 0.259·5-s + (0.325 + 0.880i)6-s + 1.57i·7-s + (0.488 − 0.872i)8-s + 0.119·9-s + (0.243 − 0.0898i)10-s + 0.676i·11-s + (0.609 + 0.713i)12-s + 1.31·13-s + (0.546 + 1.48i)14-s + 0.243i·15-s + (0.155 − 0.987i)16-s − 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(10.1405\)
Root analytic conductor: \(3.18442\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :10),\ 0.760 - 0.649i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(3.03068 + 1.11883i\)
\(L(\frac12)\) \(\approx\) \(3.03068 + 1.11883i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-960. + 354. i)T \)
good3 \( 1 - 5.54e4iT - 3.48e9T^{2} \)
5 \( 1 - 2.53e6T + 9.53e13T^{2} \)
7 \( 1 - 4.45e8iT - 7.97e16T^{2} \)
11 \( 1 - 1.75e10iT - 6.72e20T^{2} \)
13 \( 1 - 1.80e11T + 1.90e22T^{2} \)
17 \( 1 + 2.03e12T + 4.06e24T^{2} \)
19 \( 1 + 3.24e12iT - 3.75e25T^{2} \)
23 \( 1 + 4.86e13iT - 1.71e27T^{2} \)
29 \( 1 + 3.34e14T + 1.76e29T^{2} \)
31 \( 1 + 1.26e15iT - 6.71e29T^{2} \)
37 \( 1 - 5.85e15T + 2.31e31T^{2} \)
41 \( 1 - 1.81e16T + 1.80e32T^{2} \)
43 \( 1 + 1.47e16iT - 4.67e32T^{2} \)
47 \( 1 - 2.40e16iT - 2.76e33T^{2} \)
53 \( 1 + 8.83e16T + 3.05e34T^{2} \)
59 \( 1 - 1.22e17iT - 2.61e35T^{2} \)
61 \( 1 + 1.58e17T + 5.08e35T^{2} \)
67 \( 1 + 1.21e18iT - 3.32e36T^{2} \)
71 \( 1 - 4.27e18iT - 1.05e37T^{2} \)
73 \( 1 - 1.22e18T + 1.84e37T^{2} \)
79 \( 1 - 7.19e18iT - 8.96e37T^{2} \)
83 \( 1 + 2.08e19iT - 2.40e38T^{2} \)
89 \( 1 + 4.22e18T + 9.72e38T^{2} \)
97 \( 1 - 9.17e19T + 5.43e39T^{2} \)
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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

−20.55023781601364493395866547138, −18.51685119582909926254595600794, −15.85110600462846034309524129945, −15.04629925477112585935118802051, −12.95068384137070974440596755176, −11.18940395761223922480368857700, −9.381539837606128392098502472662, −5.95822293416849051997151171790, −4.30497877868084942486272405834, −2.27197674551396385180125542940, 1.37907043422237470080436202607, 3.85466356873615136407291933130, 6.30821097198180104262487828597, 7.68342790793411374249385480289, 11.06945918757812373635777320320, 13.17867250840254085327378301955, 13.82544805792769634860831616450, 16.12894476053087159239753599567, 17.71768614044567834319412339729, 19.78185736103314504111072833135

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