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Properties

Label 2-5-5.3-c32-0-2
Degree $2$
Conductor $5$
Sign $-0.0454 + 0.998i$
Analytic cond. $32.4333$
Root an. cond. $5.69502$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.79e4 + 3.79e4i)2-s + (3.04e7 + 3.04e7i)3-s + 1.41e9i·4-s + (7.42e10 + 1.33e11i)5-s − 2.31e12·6-s + (−9.26e12 + 9.26e12i)7-s + (−2.16e14 − 2.16e14i)8-s − 9.78e11i·9-s + (−7.88e15 − 2.24e15i)10-s + 1.98e16·11-s + (−4.29e16 + 4.29e16i)12-s + (−8.93e17 − 8.93e17i)13-s − 7.03e17i·14-s + (−1.79e18 + 6.31e18i)15-s + 1.03e19·16-s + (−3.66e19 + 3.66e19i)17-s + ⋯
L(s)  = 1  + (−0.579 + 0.579i)2-s + (0.706 + 0.706i)3-s + 0.328i·4-s + (0.486 + 0.873i)5-s − 0.819·6-s + (−0.278 + 0.278i)7-s + (−0.769 − 0.769i)8-s − 0.000527i·9-s + (−0.788 − 0.224i)10-s + 0.432·11-s + (−0.232 + 0.232i)12-s + (−1.34 − 1.34i)13-s − 0.323i·14-s + (−0.273 + 0.961i)15-s + 0.563·16-s + (−0.753 + 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0454 + 0.998i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0454 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.0454 + 0.998i$
Analytic conductor: \(32.4333\)
Root analytic conductor: \(5.69502\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.0454 + 0.998i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.4696101500\)
\(L(\frac12)\) \(\approx\) \(0.4696101500\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.42e10 - 1.33e11i)T \)
good2 \( 1 + (3.79e4 - 3.79e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (-3.04e7 - 3.04e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (9.26e12 - 9.26e12i)T - 1.10e27iT^{2} \)
11 \( 1 - 1.98e16T + 2.11e33T^{2} \)
13 \( 1 + (8.93e17 + 8.93e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (3.66e19 - 3.66e19i)T - 2.36e39iT^{2} \)
19 \( 1 - 2.80e20iT - 8.31e40T^{2} \)
23 \( 1 + (2.06e21 + 2.06e21i)T + 3.76e43iT^{2} \)
29 \( 1 + 1.02e23iT - 6.26e46T^{2} \)
31 \( 1 + 9.64e23T + 5.29e47T^{2} \)
37 \( 1 + (-7.27e24 + 7.27e24i)T - 1.52e50iT^{2} \)
41 \( 1 - 1.58e25T + 4.06e51T^{2} \)
43 \( 1 + (9.21e25 + 9.21e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (3.74e26 - 3.74e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (-4.43e27 - 4.43e27i)T + 1.50e55iT^{2} \)
59 \( 1 + 4.27e27iT - 4.64e56T^{2} \)
61 \( 1 + 5.42e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.45e29 + 1.45e29i)T - 2.71e58iT^{2} \)
71 \( 1 + 1.17e29T + 1.73e59T^{2} \)
73 \( 1 + (8.02e28 + 8.02e28i)T + 4.22e59iT^{2} \)
79 \( 1 - 2.05e30iT - 5.29e60T^{2} \)
83 \( 1 + (4.06e30 + 4.06e30i)T + 2.57e61iT^{2} \)
89 \( 1 - 1.23e31iT - 2.40e62T^{2} \)
97 \( 1 + (-2.90e31 + 2.90e31i)T - 3.77e63iT^{2} \)
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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

−17.09789632801578234022045131066, −15.42260040369571149221334503298, −14.61225815535622945122503493555, −12.54988592898841257805497664619, −10.21730624924337132877049796568, −9.174360082856701450177340534683, −7.68779647052863845998996514924, −6.15035492681996206929515258446, −3.70976761500393408280920809757, −2.58262807734597938689910146040, 0.14750831446699077080064432824, 1.56897917611106422276447025307, 2.40009008444455147743097926826, 4.89401115117803779520774662122, 6.94436424055258988289756763325, 8.824253368084103420529460706288, 9.671210567413512533799918881743, 11.65432421536946885083130403043, 13.27859911620536849366686318460, 14.44378499498831624130993976672

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