Properties

Label 2-2e2-4.3-c34-0-13
Degree $2$
Conductor $4$
Sign $-0.980 - 0.198i$
Analytic cond. $29.2902$
Root an. cond. $5.41204$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30e5 − 1.31e4i)2-s − 1.61e8i·3-s + (1.68e10 + 3.41e9i)4-s − 2.13e10·5-s + (−2.11e12 + 2.10e13i)6-s − 7.96e13i·7-s + (−2.15e15 − 6.66e14i)8-s − 9.33e15·9-s + (2.78e15 + 2.80e14i)10-s − 6.81e17i·11-s + (5.51e17 − 2.71e18i)12-s + 4.12e18·13-s + (−1.04e18 + 1.03e19i)14-s + 3.45e18i·15-s + (2.71e20 + 1.15e20i)16-s + 9.62e20·17-s + ⋯
L(s)  = 1  + (−0.994 − 0.0999i)2-s − 1.24i·3-s + (0.980 + 0.198i)4-s − 0.0280·5-s + (−0.124 + 1.24i)6-s − 0.342i·7-s + (−0.955 − 0.295i)8-s − 0.559·9-s + (0.0278 + 0.00280i)10-s − 1.34i·11-s + (0.248 − 1.22i)12-s + 0.476·13-s + (−0.0342 + 0.340i)14-s + 0.0350i·15-s + (0.920 + 0.389i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.980 - 0.198i$
Analytic conductor: \(29.2902\)
Root analytic conductor: \(5.41204\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :17),\ -0.980 - 0.198i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.9407776344\)
\(L(\frac12)\) \(\approx\) \(0.9407776344\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30e5 + 1.31e4i)T \)
good3 \( 1 + 1.61e8iT - 1.66e16T^{2} \)
5 \( 1 + 2.13e10T + 5.82e23T^{2} \)
7 \( 1 + 7.96e13iT - 5.41e28T^{2} \)
11 \( 1 + 6.81e17iT - 2.55e35T^{2} \)
13 \( 1 - 4.12e18T + 7.48e37T^{2} \)
17 \( 1 - 9.62e20T + 6.84e41T^{2} \)
19 \( 1 + 8.20e21iT - 3.00e43T^{2} \)
23 \( 1 + 1.41e23iT - 1.98e46T^{2} \)
29 \( 1 - 4.99e23T + 5.26e49T^{2} \)
31 \( 1 - 3.80e25iT - 5.08e50T^{2} \)
37 \( 1 + 5.54e26T + 2.08e53T^{2} \)
41 \( 1 - 1.30e27T + 6.83e54T^{2} \)
43 \( 1 + 1.77e27iT - 3.45e55T^{2} \)
47 \( 1 - 3.99e28iT - 7.10e56T^{2} \)
53 \( 1 + 1.11e29T + 4.22e58T^{2} \)
59 \( 1 - 1.19e30iT - 1.61e60T^{2} \)
61 \( 1 + 3.77e30T + 5.02e60T^{2} \)
67 \( 1 + 3.98e29iT - 1.22e62T^{2} \)
71 \( 1 + 3.35e31iT - 8.76e62T^{2} \)
73 \( 1 + 4.87e31T + 2.25e63T^{2} \)
79 \( 1 - 1.05e32iT - 3.30e64T^{2} \)
83 \( 1 + 3.41e32iT - 1.77e65T^{2} \)
89 \( 1 - 1.98e33T + 1.90e66T^{2} \)
97 \( 1 - 3.77e33T + 3.55e67T^{2} \)
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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

−16.06713704704464352048430842689, −13.78107644768556660437768290138, −12.20030615358816579175587670177, −10.74761475863610237394570324353, −8.700110026331062443886412229912, −7.44333918466676168444276130222, −6.18372333192785312332269846838, −3.00372189129651238580688133133, −1.34949165147225555236872818920, −0.43989850130560666229749287241, 1.73329452346760934239077206923, 3.70615376850611301723060785579, 5.63253276327931777494624951869, 7.72245603700836967497742055221, 9.477688392679845925282558681236, 10.25093439885253629434062399090, 11.96172402513236564984692959901, 14.86711439986350767094691313241, 15.83697385878020721541991785606, 17.12090620992914696520487374577

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