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Properties

Label 2-5-5.2-c32-0-14
Degree $2$
Conductor $5$
Sign $-0.998 - 0.0453i$
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  + (2.55e4 + 2.55e4i)2-s + (3.82e7 − 3.82e7i)3-s − 2.99e9i·4-s + (−1.26e11 − 8.60e10i)5-s + 1.95e12·6-s + (−1.22e13 − 1.22e13i)7-s + (1.85e14 − 1.85e14i)8-s − 1.06e15i·9-s + (−1.02e15 − 5.41e15i)10-s − 6.03e16·11-s + (−1.14e17 − 1.14e17i)12-s + (1.61e17 − 1.61e17i)13-s − 6.25e17i·14-s + (−8.10e18 + 1.52e18i)15-s − 3.35e18·16-s + (3.87e19 + 3.87e19i)17-s + ⋯
L(s)  = 1  + (0.389 + 0.389i)2-s + (0.888 − 0.888i)3-s − 0.696i·4-s + (−0.825 − 0.563i)5-s + 0.691·6-s + (−0.368 − 0.368i)7-s + (0.660 − 0.660i)8-s − 0.577i·9-s + (−0.102 − 0.541i)10-s − 1.31·11-s + (−0.618 − 0.618i)12-s + (0.243 − 0.243i)13-s − 0.287i·14-s + (−1.23 + 0.232i)15-s − 0.181·16-s + (0.795 + 0.795i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0453i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.998 - 0.0453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.998 - 0.0453i$
Analytic conductor: \(32.4333\)
Root analytic conductor: \(5.69502\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.998 - 0.0453i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.495088907\)
\(L(\frac12)\) \(\approx\) \(1.495088907\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.26e11 + 8.60e10i)T \)
good2 \( 1 + (-2.55e4 - 2.55e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-3.82e7 + 3.82e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (1.22e13 + 1.22e13i)T + 1.10e27iT^{2} \)
11 \( 1 + 6.03e16T + 2.11e33T^{2} \)
13 \( 1 + (-1.61e17 + 1.61e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (-3.87e19 - 3.87e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 2.45e20iT - 8.31e40T^{2} \)
23 \( 1 + (9.22e19 - 9.22e19i)T - 3.76e43iT^{2} \)
29 \( 1 + 1.82e23iT - 6.26e46T^{2} \)
31 \( 1 + 3.83e23T + 5.29e47T^{2} \)
37 \( 1 + (1.31e25 + 1.31e25i)T + 1.52e50iT^{2} \)
41 \( 1 + 1.00e26T + 4.06e51T^{2} \)
43 \( 1 + (-7.50e25 + 7.50e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (3.79e24 + 3.79e24i)T + 3.21e53iT^{2} \)
53 \( 1 + (-5.28e27 + 5.28e27i)T - 1.50e55iT^{2} \)
59 \( 1 + 2.33e28iT - 4.64e56T^{2} \)
61 \( 1 + 6.44e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.20e29 - 1.20e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 7.66e29T + 1.73e59T^{2} \)
73 \( 1 + (5.26e28 - 5.26e28i)T - 4.22e59iT^{2} \)
79 \( 1 - 2.15e30iT - 5.29e60T^{2} \)
83 \( 1 + (-4.04e30 + 4.04e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 2.02e31iT - 2.40e62T^{2} \)
97 \( 1 + (5.39e31 + 5.39e31i)T + 3.77e63iT^{2} \)
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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

−15.02434910692658140117464831940, −13.61699419634360049711317818141, −12.65058590764942550471976900414, −10.35505430781497544408030238643, −8.296063649483917692680688105322, −7.26680143945786234776988095300, −5.43145356123237274304375436774, −3.62698846605764620845717315247, −1.72574461470115892145713253948, −0.34223529706250447043109496338, 2.76994954637875909209066819208, 3.30092195277662173019817195556, 4.71686557324430808167621311202, 7.46075116898714072971828304842, 8.776380474117780569497173075236, 10.54828203599563947509318072092, 12.07567384417180509558085440067, 13.70412852381736608801471433402, 15.24646879326057665261383966725, 16.19615129973669585871067776138

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