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Properties

Label 2-2e2-4.3-c8-0-0
Degree $2$
Conductor $4$
Sign $-0.218 - 0.975i$
Analytic cond. $1.62951$
Root an. cond. $1.27652$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10 + 12.4i)2-s + 99.9i·3-s + (−56 − 249. i)4-s + 610·5-s + (−1.24e3 − 999. i)6-s + 1.39e3i·7-s + (3.68e3 + 1.79e3i)8-s − 3.42e3·9-s + (−6.10e3 + 7.61e3i)10-s − 1.84e4i·11-s + (2.49e4 − 5.59e3i)12-s − 5.47e3·13-s + (−1.74e4 − 1.39e4i)14-s + 6.09e4i·15-s + (−5.92e4 + 2.79e4i)16-s + 7.30e4·17-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)2-s + 1.23i·3-s + (−0.218 − 0.975i)4-s + 0.976·5-s + (−0.962 − 0.770i)6-s + 0.582i·7-s + (0.898 + 0.439i)8-s − 0.521·9-s + (−0.609 + 0.761i)10-s − 1.26i·11-s + (1.20 − 0.269i)12-s − 0.191·13-s + (−0.454 − 0.364i)14-s + 1.20i·15-s + (−0.904 + 0.426i)16-s + 0.875·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.218 - 0.975i$
Analytic conductor: \(1.62951\)
Root analytic conductor: \(1.27652\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :4),\ -0.218 - 0.975i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.608015 + 0.759411i\)
\(L(\frac12)\) \(\approx\) \(0.608015 + 0.759411i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10 - 12.4i)T \)
good3 \( 1 - 99.9iT - 6.56e3T^{2} \)
5 \( 1 - 610T + 3.90e5T^{2} \)
7 \( 1 - 1.39e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.84e4iT - 2.14e8T^{2} \)
13 \( 1 + 5.47e3T + 8.15e8T^{2} \)
17 \( 1 - 7.30e4T + 6.97e9T^{2} \)
19 \( 1 + 1.94e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.37e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.28e5T + 5.00e11T^{2} \)
31 \( 1 + 6.79e4iT - 8.52e11T^{2} \)
37 \( 1 + 3.47e6T + 3.51e12T^{2} \)
41 \( 1 - 2.14e6T + 7.98e12T^{2} \)
43 \( 1 + 5.92e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.62e6iT - 2.38e13T^{2} \)
53 \( 1 - 8.24e5T + 6.22e13T^{2} \)
59 \( 1 + 3.72e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.47e7T + 1.91e14T^{2} \)
67 \( 1 - 1.52e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.19e6iT - 6.45e14T^{2} \)
73 \( 1 + 5.72e6T + 8.06e14T^{2} \)
79 \( 1 - 3.59e7iT - 1.51e15T^{2} \)
83 \( 1 + 5.19e7iT - 2.25e15T^{2} \)
89 \( 1 + 8.33e7T + 3.93e15T^{2} \)
97 \( 1 - 1.20e8T + 7.83e15T^{2} \)
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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

−24.38655660422666686374795816593, −22.24342160857562206932762835749, −21.04356257984729831813228989909, −18.79159807457104404394184152711, −17.02219167945562673525700051124, −15.77275031872313453029833508855, −14.18255958717057463814967712833, −10.42394208697097118220815736412, −9.012485997895990053334791366615, −5.61132311738355574390567068510, 1.67164860332379608344561487750, 7.36865938776434711337216230376, 9.931355639451609166898450403277, 12.33245402130425562647369855681, 13.63807897941524328856721670403, 17.23027320724048646915557601166, 18.15585329877628661872716835672, 19.64969778921455067094912119585, 21.15962427867806757942742231853, 23.05283787075017977522675009153

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