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Properties

Label 2-2e2-4.3-c30-0-0
Degree $2$
Conductor $4$
Sign $0.999 - 0.0435i$
Analytic cond. $22.8057$
Root an. cond. $4.77553$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−713. − 3.27e4i)2-s − 1.39e7i·3-s + (−1.07e9 + 4.67e7i)4-s − 5.42e10·5-s + (−4.55e11 + 9.92e9i)6-s + 5.92e11i·7-s + (2.29e12 + 3.51e13i)8-s + 1.26e13·9-s + (3.87e13 + 1.77e15i)10-s − 3.23e15i·11-s + (6.50e14 + 1.49e16i)12-s − 2.32e16·13-s + (1.94e16 − 4.23e14i)14-s + 7.54e17i·15-s + (1.14e18 − 1.00e17i)16-s − 3.10e18·17-s + ⋯
L(s)  = 1  + (−0.0217 − 0.999i)2-s − 0.968i·3-s + (−0.999 + 0.0435i)4-s − 1.77·5-s + (−0.968 + 0.0211i)6-s + 0.124i·7-s + (0.0653 + 0.997i)8-s + 0.0613·9-s + (0.0387 + 1.77i)10-s − 0.774i·11-s + (0.0422 + 0.967i)12-s − 0.453·13-s + (0.124 − 0.00272i)14-s + 1.72i·15-s + (0.996 − 0.0870i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0435i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (0.999 - 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.999 - 0.0435i$
Analytic conductor: \(22.8057\)
Root analytic conductor: \(4.77553\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :15),\ 0.999 - 0.0435i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(0.297720 + 0.00648811i\)
\(L(\frac12)\) \(\approx\) \(0.297720 + 0.00648811i\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (713. + 3.27e4i)T \)
good3 \( 1 + 1.39e7iT - 2.05e14T^{2} \)
5 \( 1 + 5.42e10T + 9.31e20T^{2} \)
7 \( 1 - 5.92e11iT - 2.25e25T^{2} \)
11 \( 1 + 3.23e15iT - 1.74e31T^{2} \)
13 \( 1 + 2.32e16T + 2.61e33T^{2} \)
17 \( 1 + 3.10e18T + 8.19e36T^{2} \)
19 \( 1 + 2.48e19iT - 2.30e38T^{2} \)
23 \( 1 - 3.89e20iT - 7.10e40T^{2} \)
29 \( 1 - 8.64e21T + 7.44e43T^{2} \)
31 \( 1 - 1.93e22iT - 5.50e44T^{2} \)
37 \( 1 + 7.61e22T + 1.11e47T^{2} \)
41 \( 1 + 6.89e23T + 2.41e48T^{2} \)
43 \( 1 - 2.04e24iT - 1.00e49T^{2} \)
47 \( 1 - 1.46e25iT - 1.45e50T^{2} \)
53 \( 1 - 3.50e25T + 5.34e51T^{2} \)
59 \( 1 - 4.94e26iT - 1.33e53T^{2} \)
61 \( 1 + 2.04e26T + 3.62e53T^{2} \)
67 \( 1 + 3.08e26iT - 6.05e54T^{2} \)
71 \( 1 - 9.39e26iT - 3.44e55T^{2} \)
73 \( 1 + 1.34e28T + 7.93e55T^{2} \)
79 \( 1 + 2.95e28iT - 8.48e56T^{2} \)
83 \( 1 - 6.94e28iT - 3.73e57T^{2} \)
89 \( 1 + 1.83e29T + 3.03e58T^{2} \)
97 \( 1 + 5.60e29T + 4.01e59T^{2} \)
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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

−17.87037157063412405359473411508, −15.56557726415269762217548014820, −13.44805749532787508026639989794, −12.11295165699612599521924709742, −11.15458223255676457544407790786, −8.653228710812344996180816456995, −7.27445407322411151638369260754, −4.46422199547042798294418922452, −2.89696560593351002492719260762, −0.977215028155075648973096024427, 0.13945143904763071298612477702, 3.90788616496938875189245220985, 4.62757568878527318163202826082, 7.08459263132186899324907317714, 8.448276413341838976067544678752, 10.26434030613254302701312010028, 12.34260788883603575298212881931, 14.79364333601345493193086882249, 15.61025021086146561152742381651, 16.64339572433870894118046419290

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