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Properties

Label 2-5-5.4-c25-0-10
Degree $2$
Conductor $5$
Sign $-0.150 + 0.988i$
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  + 2.02e3i·2-s − 5.29e5i·3-s + 2.94e7·4-s + (8.23e7 − 5.39e8i)5-s + 1.07e9·6-s − 2.78e10i·7-s + 1.27e11i·8-s + 5.66e11·9-s + (1.09e12 + 1.66e11i)10-s − 1.72e13·11-s − 1.56e13i·12-s + 4.19e13i·13-s + 5.63e13·14-s + (−2.85e14 − 4.36e13i)15-s + 7.30e14·16-s − 2.71e15i·17-s + ⋯
L(s)  = 1  + 0.349i·2-s − 0.575i·3-s + 0.878·4-s + (0.150 − 0.988i)5-s + 0.201·6-s − 0.760i·7-s + 0.655i·8-s + 0.668·9-s + (0.345 + 0.0527i)10-s − 1.65·11-s − 0.505i·12-s + 0.499i·13-s + 0.265·14-s + (−0.568 − 0.0868i)15-s + 0.648·16-s − 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.150 + 0.988i$
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: \((2,\ 5,\ (\ :25/2),\ -0.150 + 0.988i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.34837 - 1.56983i\)
\(L(\frac12)\) \(\approx\) \(1.34837 - 1.56983i\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-8.23e7 + 5.39e8i)T \)
good2 \( 1 - 2.02e3iT - 3.35e7T^{2} \)
3 \( 1 + 5.29e5iT - 8.47e11T^{2} \)
7 \( 1 + 2.78e10iT - 1.34e21T^{2} \)
11 \( 1 + 1.72e13T + 1.08e26T^{2} \)
13 \( 1 - 4.19e13iT - 7.05e27T^{2} \)
17 \( 1 + 2.71e15iT - 5.77e30T^{2} \)
19 \( 1 + 2.11e15T + 9.30e31T^{2} \)
23 \( 1 + 1.88e17iT - 1.10e34T^{2} \)
29 \( 1 - 1.94e18T + 3.63e36T^{2} \)
31 \( 1 + 3.33e18T + 1.92e37T^{2} \)
37 \( 1 - 6.23e19iT - 1.60e39T^{2} \)
41 \( 1 + 5.06e19T + 2.08e40T^{2} \)
43 \( 1 + 2.68e20iT - 6.86e40T^{2} \)
47 \( 1 + 1.57e20iT - 6.34e41T^{2} \)
53 \( 1 - 1.18e19iT - 1.27e43T^{2} \)
59 \( 1 - 2.75e21T + 1.86e44T^{2} \)
61 \( 1 + 4.94e21T + 4.29e44T^{2} \)
67 \( 1 + 4.65e22iT - 4.48e45T^{2} \)
71 \( 1 - 1.36e23T + 1.91e46T^{2} \)
73 \( 1 - 4.38e22iT - 3.82e46T^{2} \)
79 \( 1 - 6.86e23T + 2.75e47T^{2} \)
83 \( 1 - 1.11e24iT - 9.48e47T^{2} \)
89 \( 1 + 8.75e23T + 5.42e48T^{2} \)
97 \( 1 - 8.28e24iT - 4.66e49T^{2} \)
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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

−16.72762848192603282289866666407, −15.75564079849883174377282136134, −13.59931202880355798795823296643, −12.29909165818471203760652095714, −10.38699776030346356902337536828, −8.054969517950303876483670312078, −6.80779005563363473915970318914, −4.88614547941485529442903507396, −2.26688796256954742390729287060, −0.71214321673597195099976912568, 2.04963738613577035186767834951, 3.32324998553507603714917581391, 5.73805219857841068316205032954, 7.53563696941935960875573616328, 10.06981714436225394013914119980, 10.96084246391001188424638496505, 12.80049475515620133414406314034, 15.17248412572866820846926835119, 15.78390084221511270633011635904, 18.08253048929777100229281420760

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