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Properties

Label 2-2e2-4.3-c38-0-13
Degree $2$
Conductor $4$
Sign $-0.977 - 0.212i$
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  + (−5.21e5 − 5.60e4i)2-s − 1.48e9i·3-s + (2.68e11 + 5.84e10i)4-s + 1.90e13·5-s + (−8.31e13 + 7.72e14i)6-s + 1.45e16i·7-s + (−1.36e17 − 4.55e16i)8-s − 8.46e17·9-s + (−9.91e18 − 1.06e18i)10-s + 3.87e19i·11-s + (8.66e19 − 3.98e20i)12-s − 8.59e20·13-s + (8.14e20 − 7.57e21i)14-s − 2.81e22i·15-s + (6.87e22 + 3.14e22i)16-s − 1.60e23·17-s + ⋯
L(s)  = 1  + (−0.994 − 0.106i)2-s − 1.27i·3-s + (0.977 + 0.212i)4-s + 0.997·5-s + (−0.136 + 1.26i)6-s + 1.27i·7-s + (−0.948 − 0.316i)8-s − 0.626·9-s + (−0.991 − 0.106i)10-s + 0.633i·11-s + (0.271 − 1.24i)12-s − 0.588·13-s + (0.136 − 1.26i)14-s − 1.27i·15-s + (0.909 + 0.415i)16-s − 0.670·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.977 - 0.212i$
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: \((2,\ 4,\ (\ :19),\ -0.977 - 0.212i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.4057277950\)
\(L(\frac12)\) \(\approx\) \(0.4057277950\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.21e5 + 5.60e4i)T \)
good3 \( 1 + 1.48e9iT - 1.35e18T^{2} \)
5 \( 1 - 1.90e13T + 3.63e26T^{2} \)
7 \( 1 - 1.45e16iT - 1.29e32T^{2} \)
11 \( 1 - 3.87e19iT - 3.74e39T^{2} \)
13 \( 1 + 8.59e20T + 2.13e42T^{2} \)
17 \( 1 + 1.60e23T + 5.71e46T^{2} \)
19 \( 1 - 1.01e24iT - 3.91e48T^{2} \)
23 \( 1 + 1.41e26iT - 5.56e51T^{2} \)
29 \( 1 + 9.93e27T + 3.72e55T^{2} \)
31 \( 1 + 3.69e28iT - 4.69e56T^{2} \)
37 \( 1 + 7.42e29T + 3.90e59T^{2} \)
41 \( 1 + 6.52e30T + 1.93e61T^{2} \)
43 \( 1 - 5.95e30iT - 1.17e62T^{2} \)
47 \( 1 + 9.29e31iT - 3.46e63T^{2} \)
53 \( 1 - 3.86e32T + 3.33e65T^{2} \)
59 \( 1 + 2.63e33iT - 1.96e67T^{2} \)
61 \( 1 - 2.89e33T + 6.95e67T^{2} \)
67 \( 1 - 2.95e34iT - 2.45e69T^{2} \)
71 \( 1 - 1.32e35iT - 2.22e70T^{2} \)
73 \( 1 + 1.42e34T + 6.40e70T^{2} \)
79 \( 1 + 5.08e35iT - 1.28e72T^{2} \)
83 \( 1 + 2.79e36iT - 8.41e72T^{2} \)
89 \( 1 + 4.53e36T + 1.19e74T^{2} \)
97 \( 1 + 9.50e37T + 3.14e75T^{2} \)
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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

−15.00434600839345257436828377836, −12.95321495253187490306499671062, −11.90718769251672145794214107525, −9.878018898647202509723618026877, −8.522335491018788049258039821413, −6.98198808000884581317179353427, −5.83900066069711302075381766438, −2.20934612109603256852125133126, −1.98015628807142485968789618107, −0.14987292968830369428384423052, 1.55743890992177442311076317438, 3.46919360136346289592478701681, 5.33638967377051845968389405237, 7.12580194072608358911560717388, 9.134921221393867378428369822405, 10.08411202788642386777905574493, 11.02499485072578229797586591109, 13.77859102862294456715446789496, 15.45291872487586377262027535676, 16.75481847735813480273652289582

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