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Properties

Label 2-2e2-4.3-c22-0-5
Degree $2$
Conductor $4$
Sign $-0.517 - 0.855i$
Analytic cond. $12.2682$
Root an. cond. $3.50261$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.78e3 + 1.00e3i)2-s + 2.67e5i·3-s + (2.16e6 + 3.59e6i)4-s + 8.57e7·5-s + (−2.69e8 + 4.77e8i)6-s − 2.80e8i·7-s + (2.54e8 + 8.58e9i)8-s − 4.03e10·9-s + (1.53e11 + 8.63e10i)10-s − 3.14e11i·11-s + (−9.61e11 + 5.80e11i)12-s − 8.66e11·13-s + (2.82e11 − 5.00e11i)14-s + 2.29e13i·15-s + (−8.18e12 + 1.55e13i)16-s − 3.75e12·17-s + ⋯
L(s)  = 1  + (0.870 + 0.491i)2-s + 1.51i·3-s + (0.517 + 0.855i)4-s + 1.75·5-s + (−0.743 + 1.31i)6-s − 0.141i·7-s + (0.0296 + 0.999i)8-s − 1.28·9-s + (1.53 + 0.863i)10-s − 1.10i·11-s + (−1.29 + 0.781i)12-s − 0.483·13-s + (0.0697 − 0.123i)14-s + 2.65i·15-s + (−0.465 + 0.885i)16-s − 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.517 - 0.855i$
Analytic conductor: \(12.2682\)
Root analytic conductor: \(3.50261\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :11),\ -0.517 - 0.855i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(1.80518 + 3.19924i\)
\(L(\frac12)\) \(\approx\) \(1.80518 + 3.19924i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78e3 - 1.00e3i)T \)
good3 \( 1 - 2.67e5iT - 3.13e10T^{2} \)
5 \( 1 - 8.57e7T + 2.38e15T^{2} \)
7 \( 1 + 2.80e8iT - 3.90e18T^{2} \)
11 \( 1 + 3.14e11iT - 8.14e22T^{2} \)
13 \( 1 + 8.66e11T + 3.21e24T^{2} \)
17 \( 1 + 3.75e12T + 1.17e27T^{2} \)
19 \( 1 + 1.33e14iT - 1.35e28T^{2} \)
23 \( 1 + 6.25e14iT - 9.07e29T^{2} \)
29 \( 1 + 8.54e14T + 1.48e32T^{2} \)
31 \( 1 + 1.29e16iT - 6.45e32T^{2} \)
37 \( 1 - 7.84e16T + 3.16e34T^{2} \)
41 \( 1 + 1.08e17T + 3.02e35T^{2} \)
43 \( 1 - 5.77e17iT - 8.63e35T^{2} \)
47 \( 1 - 3.78e18iT - 6.11e36T^{2} \)
53 \( 1 + 1.21e19T + 8.59e37T^{2} \)
59 \( 1 - 3.48e19iT - 9.09e38T^{2} \)
61 \( 1 - 3.03e19T + 1.89e39T^{2} \)
67 \( 1 + 9.42e19iT - 1.49e40T^{2} \)
71 \( 1 + 2.75e20iT - 5.34e40T^{2} \)
73 \( 1 + 4.18e20T + 9.84e40T^{2} \)
79 \( 1 + 9.99e20iT - 5.59e41T^{2} \)
83 \( 1 - 1.27e20iT - 1.65e42T^{2} \)
89 \( 1 - 5.20e21T + 7.70e42T^{2} \)
97 \( 1 + 2.44e21T + 5.11e43T^{2} \)
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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

−20.65418610126768238869803894662, −17.35518077947552918867902427076, −16.28397300846435486785999170332, −14.65739225337866038342381442118, −13.44236073792387832855622268851, −10.79460263015245135932372863110, −9.184583881339319508705013882059, −6.04322856686263116102633392354, −4.76100766037343850350404864441, −2.80404887574907167836000074399, 1.49285589706981910240971671719, 2.27721044124581128629162492531, 5.54839421007273042920232365543, 6.86843913823520686296416546275, 9.922381899759808593298093945179, 12.28546431028286613487571872425, 13.28841599131343574019751678818, 14.41811492188036578759804190983, 17.47421597377568245791385659065, 18.67207661346033058765992552524

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