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Properties

Label 2-2e2-4.3-c24-0-1
Degree $2$
Conductor $4$
Sign $-0.438 + 0.898i$
Analytic cond. $14.5986$
Root an. cond. $3.82082$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.16e3 + 3.47e3i)2-s + 5.69e5i·3-s + (−7.36e6 + 1.50e7i)4-s − 2.26e8·5-s + (−1.97e9 + 1.23e9i)6-s − 6.45e9i·7-s + (−6.83e10 + 7.12e9i)8-s − 4.14e10·9-s + (−4.91e11 − 7.87e11i)10-s + 2.08e12i·11-s + (−8.57e12 − 4.19e12i)12-s + 2.81e13·13-s + (2.24e13 − 1.39e13i)14-s − 1.29e14i·15-s + (−1.73e14 − 2.22e14i)16-s − 1.12e15·17-s + ⋯
L(s)  = 1  + (0.529 + 0.848i)2-s + 1.07i·3-s + (−0.438 + 0.898i)4-s − 0.928·5-s + (−0.908 + 0.567i)6-s − 0.466i·7-s + (−0.994 + 0.103i)8-s − 0.146·9-s + (−0.491 − 0.787i)10-s + 0.662i·11-s + (−0.962 − 0.469i)12-s + 1.20·13-s + (0.395 − 0.246i)14-s − 0.994i·15-s + (−0.614 − 0.788i)16-s − 1.92·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.438 + 0.898i$
Analytic conductor: \(14.5986\)
Root analytic conductor: \(3.82082\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :12),\ -0.438 + 0.898i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.508259 - 0.813898i\)
\(L(\frac12)\) \(\approx\) \(0.508259 - 0.813898i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.16e3 - 3.47e3i)T \)
good3 \( 1 - 5.69e5iT - 2.82e11T^{2} \)
5 \( 1 + 2.26e8T + 5.96e16T^{2} \)
7 \( 1 + 6.45e9iT - 1.91e20T^{2} \)
11 \( 1 - 2.08e12iT - 9.84e24T^{2} \)
13 \( 1 - 2.81e13T + 5.42e26T^{2} \)
17 \( 1 + 1.12e15T + 3.39e29T^{2} \)
19 \( 1 + 3.31e15iT - 4.89e30T^{2} \)
23 \( 1 - 2.45e16iT - 4.80e32T^{2} \)
29 \( 1 + 1.79e17T + 1.25e35T^{2} \)
31 \( 1 - 8.00e17iT - 6.20e35T^{2} \)
37 \( 1 + 7.15e18T + 4.33e37T^{2} \)
41 \( 1 + 3.37e18T + 5.09e38T^{2} \)
43 \( 1 - 3.50e19iT - 1.59e39T^{2} \)
47 \( 1 + 2.96e19iT - 1.35e40T^{2} \)
53 \( 1 + 4.27e20T + 2.41e41T^{2} \)
59 \( 1 - 1.92e21iT - 3.16e42T^{2} \)
61 \( 1 - 5.04e20T + 7.04e42T^{2} \)
67 \( 1 + 2.01e21iT - 6.69e43T^{2} \)
71 \( 1 - 2.05e22iT - 2.69e44T^{2} \)
73 \( 1 - 2.34e22T + 5.24e44T^{2} \)
79 \( 1 - 3.47e22iT - 3.49e45T^{2} \)
83 \( 1 + 1.74e23iT - 1.14e46T^{2} \)
89 \( 1 - 3.70e22T + 6.10e46T^{2} \)
97 \( 1 - 4.67e22T + 4.81e47T^{2} \)
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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

−20.11034350133086481889032785434, −17.64239604245722409323249711319, −15.81313872443768718990063040996, −15.41107881159157145711061296698, −13.35003283121392044728731084102, −11.19041238370889029657685868766, −8.938360686146187459248717568412, −7.04494830655803082860883531783, −4.68423086521198200578766161801, −3.70977083464857759476211855639, 0.32647697656875805528133763271, 1.94534571192499017262710902496, 3.89484506544319212878189775075, 6.25610374069493220488036902113, 8.465885933282339500366978130109, 11.09356442610746170660283849871, 12.37004142794245600155157040252, 13.63311818486993120674859672329, 15.56891090525980394231346855836, 18.36762983282895163333068670166

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