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Properties

Label 2-5-5.4-c25-0-1
Degree $2$
Conductor $5$
Sign $-0.933 + 0.359i$
Analytic cond. $19.7998$
Root an. cond. $4.44970$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.05e4i·2-s − 1.42e6i·3-s − 7.87e7·4-s + (5.09e8 − 1.96e8i)5-s + 1.50e10·6-s + 3.45e10i·7-s − 4.78e11i·8-s − 1.17e12·9-s + (2.07e12 + 5.39e12i)10-s − 7.99e12·11-s + 1.11e14i·12-s + 2.45e13i·13-s − 3.65e14·14-s + (−2.78e14 − 7.24e14i)15-s + 2.42e15·16-s + 1.75e15i·17-s + ⋯
L(s)  = 1  + 1.82i·2-s − 1.54i·3-s − 2.34·4-s + (0.933 − 0.359i)5-s + 2.82·6-s + 0.942i·7-s − 2.46i·8-s − 1.38·9-s + (0.656 + 1.70i)10-s − 0.768·11-s + 3.62i·12-s + 0.292i·13-s − 1.72·14-s + (−0.554 − 1.44i)15-s + 2.15·16-s + 0.731i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(19.7998\)
Root analytic conductor: \(4.44970\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :25/2),\ -0.933 + 0.359i)\)

Particular Values

\(L(13)\) \(\approx\) \(0.115579 - 0.622377i\)
\(L(\frac12)\) \(\approx\) \(0.115579 - 0.622377i\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.09e8 + 1.96e8i)T \)
good2 \( 1 - 1.05e4iT - 3.35e7T^{2} \)
3 \( 1 + 1.42e6iT - 8.47e11T^{2} \)
7 \( 1 - 3.45e10iT - 1.34e21T^{2} \)
11 \( 1 + 7.99e12T + 1.08e26T^{2} \)
13 \( 1 - 2.45e13iT - 7.05e27T^{2} \)
17 \( 1 - 1.75e15iT - 5.77e30T^{2} \)
19 \( 1 + 1.10e16T + 9.30e31T^{2} \)
23 \( 1 - 1.79e17iT - 1.10e34T^{2} \)
29 \( 1 + 2.26e18T + 3.63e36T^{2} \)
31 \( 1 + 4.56e18T + 1.92e37T^{2} \)
37 \( 1 - 3.51e19iT - 1.60e39T^{2} \)
41 \( 1 + 8.43e19T + 2.08e40T^{2} \)
43 \( 1 - 5.67e19iT - 6.86e40T^{2} \)
47 \( 1 - 3.57e20iT - 6.34e41T^{2} \)
53 \( 1 - 1.71e21iT - 1.27e43T^{2} \)
59 \( 1 + 9.09e21T + 1.86e44T^{2} \)
61 \( 1 + 7.31e21T + 4.29e44T^{2} \)
67 \( 1 + 4.11e21iT - 4.48e45T^{2} \)
71 \( 1 + 5.06e21T + 1.91e46T^{2} \)
73 \( 1 + 1.13e23iT - 3.82e46T^{2} \)
79 \( 1 - 8.56e23T + 2.75e47T^{2} \)
83 \( 1 - 3.95e23iT - 9.48e47T^{2} \)
89 \( 1 - 3.25e24T + 5.42e48T^{2} \)
97 \( 1 + 5.43e24iT - 4.66e49T^{2} \)
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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

−17.96486697254944889455668478327, −16.94767444244452143581845475231, −15.11531507306522633368948953249, −13.58692475561775380504848316067, −12.75257827586347832817580982370, −9.025801984634834545409792825711, −7.77951440209307088553346273722, −6.32842536132531543392953380637, −5.45134938856798145186842033722, −1.80800136181170211119533386376, 0.22057167803767045601664999026, 2.34748511022892646025815074952, 3.72954026329191503416707860711, 5.00328332655389492605241508040, 9.160063844727082263649777052135, 10.40708450845264808257268236113, 10.76212621121413444101300451232, 13.09250374809896071420374293945, 14.51504223563172511043412206787, 16.84299212305879239501298853955

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