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Properties

Label 2-5-5.3-c32-0-7
Degree $2$
Conductor $5$
Sign $0.915 + 0.402i$
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  + (−7.09e4 + 7.09e4i)2-s + (−4.83e7 − 4.83e7i)3-s − 5.77e9i·4-s + (1.51e11 − 2.11e10i)5-s + 6.86e12·6-s + (2.09e13 − 2.09e13i)7-s + (1.05e14 + 1.05e14i)8-s + 2.82e15i·9-s + (−9.22e15 + 1.22e16i)10-s + 6.47e16·11-s + (−2.79e17 + 2.79e17i)12-s + (−4.20e17 − 4.20e17i)13-s + 2.97e18i·14-s + (−8.33e18 − 6.28e18i)15-s + 9.89e18·16-s + (−1.82e19 + 1.82e19i)17-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + (−1.12 − 1.12i)3-s − 1.34i·4-s + (0.990 − 0.138i)5-s + 2.43·6-s + (0.631 − 0.631i)7-s + (0.373 + 0.373i)8-s + 1.52i·9-s + (−0.922 + 1.22i)10-s + 1.40·11-s + (−1.51 + 1.51i)12-s + (−0.631 − 0.631i)13-s + 1.36i·14-s + (−1.26 − 0.957i)15-s + 0.536·16-s + (−0.374 + 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.915 + 0.402i$
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: \((2,\ 5,\ (\ :16),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.001949478\)
\(L(\frac12)\) \(\approx\) \(1.001949478\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.51e11 + 2.11e10i)T \)
good2 \( 1 + (7.09e4 - 7.09e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (4.83e7 + 4.83e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (-2.09e13 + 2.09e13i)T - 1.10e27iT^{2} \)
11 \( 1 - 6.47e16T + 2.11e33T^{2} \)
13 \( 1 + (4.20e17 + 4.20e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (1.82e19 - 1.82e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 1.23e20iT - 8.31e40T^{2} \)
23 \( 1 + (-6.48e21 - 6.48e21i)T + 3.76e43iT^{2} \)
29 \( 1 - 4.68e23iT - 6.26e46T^{2} \)
31 \( 1 - 2.94e22T + 5.29e47T^{2} \)
37 \( 1 + (-1.50e25 + 1.50e25i)T - 1.52e50iT^{2} \)
41 \( 1 - 4.08e25T + 4.06e51T^{2} \)
43 \( 1 + (-5.01e25 - 5.01e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (-4.23e26 + 4.23e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (2.94e27 + 2.94e27i)T + 1.50e55iT^{2} \)
59 \( 1 + 8.62e27iT - 4.64e56T^{2} \)
61 \( 1 - 1.77e28T + 1.35e57T^{2} \)
67 \( 1 + (2.48e28 - 2.48e28i)T - 2.71e58iT^{2} \)
71 \( 1 - 3.63e29T + 1.73e59T^{2} \)
73 \( 1 + (5.94e29 + 5.94e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 1.45e30iT - 5.29e60T^{2} \)
83 \( 1 + (-3.27e30 - 3.27e30i)T + 2.57e61iT^{2} \)
89 \( 1 + 8.42e29iT - 2.40e62T^{2} \)
97 \( 1 + (3.56e30 - 3.56e30i)T - 3.77e63iT^{2} \)
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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

−16.74341081844425880310305619526, −14.48848646821181333458468958081, −12.76960997434110823433200511024, −10.94349309069779896777343804498, −9.199063589683072367724594558473, −7.41222521470543879909704288652, −6.54851499935497479961433866795, −5.35322695296493929250694726968, −1.46715872775217842967639224876, −0.75024511446061930991901948390, 0.941583312007541379246396929876, 2.37272769874606198761115224424, 4.51638423015725457589907455153, 6.10809379507722257350488192729, 9.063345502983287190655223950613, 9.843228428671422998276405018928, 11.13475385201911240749680085501, 11.97359311768159133070201062428, 14.71628060802011380993905588146, 16.88362763081870737818362337525

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