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Properties

Label 2-2e2-4.3-c38-0-8
Degree $2$
Conductor $4$
Sign $-0.0704 - 0.997i$
Analytic cond. $36.5853$
Root an. cond. $6.04858$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.83e5 + 3.57e5i)2-s − 4.72e8i·3-s + (1.93e10 + 2.74e11i)4-s + 2.09e13·5-s + (1.68e14 − 1.81e14i)6-s + 1.08e16i·7-s + (−9.05e16 + 1.12e17i)8-s + 1.12e18·9-s + (8.01e18 + 7.47e18i)10-s − 5.05e19i·11-s + (1.29e20 − 9.14e18i)12-s − 7.96e20·13-s + (−3.89e21 + 4.17e21i)14-s − 9.87e21i·15-s + (−7.48e22 + 1.06e22i)16-s + 3.83e23·17-s + ⋯
L(s)  = 1  + (0.731 + 0.681i)2-s − 0.406i·3-s + (0.0704 + 0.997i)4-s + 1.09·5-s + (0.276 − 0.297i)6-s + 0.955i·7-s + (−0.628 + 0.777i)8-s + 0.834·9-s + (0.801 + 0.747i)10-s − 0.827i·11-s + (0.405 − 0.0286i)12-s − 0.544·13-s + (−0.651 + 0.698i)14-s − 0.445i·15-s + (−0.990 + 0.140i)16-s + 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0704 - 0.997i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.0704 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.0704 - 0.997i$
Analytic conductor: \(36.5853\)
Root analytic conductor: \(6.04858\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :19),\ -0.0704 - 0.997i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(3.871352170\)
\(L(\frac12)\) \(\approx\) \(3.871352170\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.83e5 - 3.57e5i)T \)
good3 \( 1 + 4.72e8iT - 1.35e18T^{2} \)
5 \( 1 - 2.09e13T + 3.63e26T^{2} \)
7 \( 1 - 1.08e16iT - 1.29e32T^{2} \)
11 \( 1 + 5.05e19iT - 3.74e39T^{2} \)
13 \( 1 + 7.96e20T + 2.13e42T^{2} \)
17 \( 1 - 3.83e23T + 5.71e46T^{2} \)
19 \( 1 - 6.36e23iT - 3.91e48T^{2} \)
23 \( 1 - 1.18e26iT - 5.56e51T^{2} \)
29 \( 1 - 1.59e27T + 3.72e55T^{2} \)
31 \( 1 - 1.54e28iT - 4.69e56T^{2} \)
37 \( 1 - 4.02e29T + 3.90e59T^{2} \)
41 \( 1 + 5.97e30T + 1.93e61T^{2} \)
43 \( 1 - 1.62e31iT - 1.17e62T^{2} \)
47 \( 1 - 7.30e31iT - 3.46e63T^{2} \)
53 \( 1 + 3.06e32T + 3.33e65T^{2} \)
59 \( 1 + 5.23e33iT - 1.96e67T^{2} \)
61 \( 1 - 8.85e33T + 6.95e67T^{2} \)
67 \( 1 + 2.65e34iT - 2.45e69T^{2} \)
71 \( 1 - 4.15e34iT - 2.22e70T^{2} \)
73 \( 1 - 1.69e35T + 6.40e70T^{2} \)
79 \( 1 + 1.90e36iT - 1.28e72T^{2} \)
83 \( 1 + 5.71e36iT - 8.41e72T^{2} \)
89 \( 1 - 1.56e36T + 1.19e74T^{2} \)
97 \( 1 - 7.60e37T + 3.14e75T^{2} \)
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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

−16.00799832247054887336691095190, −14.43552242896917282231817705057, −13.22668763100205037426806181667, −12.02209482215305252744066356967, −9.597992487981291255640748257805, −7.82873510220928886130977730986, −6.19727961868598092099841963638, −5.25751499830763088941976195093, −3.13918145449111255888887292375, −1.62806205615226233917021721143, 0.940220431571658112268052735720, 2.23719906369524724306597561972, 3.98822432203525906795945544911, 5.19773409071009309164867938672, 6.91444110794498259952134543527, 9.841010757448833544954671848301, 10.29847652582535974195684449457, 12.41922143632575159586346944805, 13.68267604711125506643319785831, 14.90458570610706751856654547560

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