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Properties

Label 2-2e2-4.3-c16-0-3
Degree $2$
Conductor $4$
Sign $0.384 + 0.923i$
Analytic cond. $6.49298$
Root an. cond. $2.54813$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−212. − 142. i)2-s + 2.90e3i·3-s + (2.51e4 + 6.05e4i)4-s + 2.08e4·5-s + (4.13e5 − 6.19e5i)6-s − 8.01e6i·7-s + (3.23e6 − 1.64e7i)8-s + 3.45e7·9-s + (−4.44e6 − 2.96e6i)10-s − 1.22e8i·11-s + (−1.75e8 + 7.32e7i)12-s + 4.11e8·13-s + (−1.13e9 + 1.70e9i)14-s + 6.07e7i·15-s + (−3.02e9 + 3.04e9i)16-s + 1.07e10·17-s + ⋯
L(s)  = 1  + (−0.831 − 0.554i)2-s + 0.443i·3-s + (0.384 + 0.923i)4-s + 0.0534·5-s + (0.245 − 0.368i)6-s − 1.39i·7-s + (0.192 − 0.981i)8-s + 0.803·9-s + (−0.0444 − 0.0296i)10-s − 0.572i·11-s + (−0.409 + 0.170i)12-s + 0.504·13-s + (−0.771 + 1.15i)14-s + 0.0236i·15-s + (−0.704 + 0.709i)16-s + 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.384 + 0.923i$
Analytic conductor: \(6.49298\)
Root analytic conductor: \(2.54813\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :8),\ 0.384 + 0.923i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.933329 - 0.622471i\)
\(L(\frac12)\) \(\approx\) \(0.933329 - 0.622471i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (212. + 142. i)T \)
good3 \( 1 - 2.90e3iT - 4.30e7T^{2} \)
5 \( 1 - 2.08e4T + 1.52e11T^{2} \)
7 \( 1 + 8.01e6iT - 3.32e13T^{2} \)
11 \( 1 + 1.22e8iT - 4.59e16T^{2} \)
13 \( 1 - 4.11e8T + 6.65e17T^{2} \)
17 \( 1 - 1.07e10T + 4.86e19T^{2} \)
19 \( 1 + 3.07e10iT - 2.88e20T^{2} \)
23 \( 1 - 6.36e10iT - 6.13e21T^{2} \)
29 \( 1 + 3.76e11T + 2.50e23T^{2} \)
31 \( 1 + 3.36e11iT - 7.27e23T^{2} \)
37 \( 1 - 1.97e12T + 1.23e25T^{2} \)
41 \( 1 - 4.08e12T + 6.37e25T^{2} \)
43 \( 1 + 1.53e13iT - 1.36e26T^{2} \)
47 \( 1 - 2.29e13iT - 5.66e26T^{2} \)
53 \( 1 + 5.35e13T + 3.87e27T^{2} \)
59 \( 1 + 6.70e13iT - 2.15e28T^{2} \)
61 \( 1 - 1.86e14T + 3.67e28T^{2} \)
67 \( 1 - 1.58e14iT - 1.64e29T^{2} \)
71 \( 1 - 8.90e14iT - 4.16e29T^{2} \)
73 \( 1 + 2.83e14T + 6.50e29T^{2} \)
79 \( 1 + 1.41e15iT - 2.30e30T^{2} \)
83 \( 1 - 2.82e15iT - 5.07e30T^{2} \)
89 \( 1 - 2.93e15T + 1.54e31T^{2} \)
97 \( 1 - 1.34e15T + 6.14e31T^{2} \)
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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

−20.54668458340273373997176401186, −19.08423024148868360831969327176, −17.32107418810611907893459990098, −15.99742649996305973905479983637, −13.35268799828013685040755498060, −11.06274008057969706697583180949, −9.701956409010880580319365874715, −7.45104044954347351979330596692, −3.75948318898384647803063157786, −0.937979288549224216735498697074, 1.68405281564334651456960302436, 5.90916813038116910575819896514, 7.935170204265300685333626305924, 9.823593637511420333097588124274, 12.24597474112844225756790115226, 14.75683063117414055357645241112, 16.24633762778350453915306930662, 18.22616154754708265382528955038, 18.92488894040954648780751556697, 21.03478925087392251057820706027

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