↑

Properties

Label 2-5-5.2-c26-0-2
Degree $2$
Conductor $5$
Sign $-0.664 - 0.747i$
Analytic cond. $21.4146$
Root an. cond. $4.62759$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8.96e3 − 8.96e3i)2-s + (−1.25e6 + 1.25e6i)3-s + 9.37e7i·4-s + (2.09e8 + 1.20e9i)5-s + 2.25e10·6-s + (5.31e10 + 5.31e10i)7-s + (2.38e11 − 2.38e11i)8-s − 6.27e11i·9-s + (8.90e12 − 1.26e13i)10-s − 1.06e13·11-s + (−1.17e14 − 1.17e14i)12-s + (1.39e14 − 1.39e14i)13-s − 9.53e14i·14-s + (−1.77e15 − 1.24e15i)15-s + 2.01e15·16-s + (9.12e15 + 9.12e15i)17-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s + (−0.789 + 0.789i)3-s + 1.39i·4-s + (0.171 + 0.985i)5-s + 1.72·6-s + (0.548 + 0.548i)7-s + (0.433 − 0.433i)8-s − 0.246i·9-s + (0.890 − 1.26i)10-s − 0.307·11-s + (−1.10 − 1.10i)12-s + (0.459 − 0.459i)13-s − 1.20i·14-s + (−0.913 − 0.642i)15-s + 0.446·16-s + (0.920 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(21.4146\)
Root analytic conductor: \(4.62759\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :13),\ -0.664 - 0.747i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.244684 + 0.544637i\)
\(L(\frac12)\) \(\approx\) \(0.244684 + 0.544637i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.09e8 - 1.20e9i)T \)
good2 \( 1 + (8.96e3 + 8.96e3i)T + 6.71e7iT^{2} \)
3 \( 1 + (1.25e6 - 1.25e6i)T - 2.54e12iT^{2} \)
7 \( 1 + (-5.31e10 - 5.31e10i)T + 9.38e21iT^{2} \)
11 \( 1 + 1.06e13T + 1.19e27T^{2} \)
13 \( 1 + (-1.39e14 + 1.39e14i)T - 9.17e28iT^{2} \)
17 \( 1 + (-9.12e15 - 9.12e15i)T + 9.81e31iT^{2} \)
19 \( 1 - 7.63e16iT - 1.76e33T^{2} \)
23 \( 1 + (-3.29e17 + 3.29e17i)T - 2.54e35iT^{2} \)
29 \( 1 - 1.91e19iT - 1.05e38T^{2} \)
31 \( 1 - 2.39e19T + 5.96e38T^{2} \)
37 \( 1 + (5.76e19 + 5.76e19i)T + 5.93e40iT^{2} \)
41 \( 1 + 1.23e21T + 8.55e41T^{2} \)
43 \( 1 + (5.39e19 - 5.39e19i)T - 2.95e42iT^{2} \)
47 \( 1 + (1.83e21 + 1.83e21i)T + 2.98e43iT^{2} \)
53 \( 1 + (1.38e22 - 1.38e22i)T - 6.77e44iT^{2} \)
59 \( 1 - 1.52e23iT - 1.10e46T^{2} \)
61 \( 1 - 7.82e22T + 2.62e46T^{2} \)
67 \( 1 + (2.38e23 + 2.38e23i)T + 3.00e47iT^{2} \)
71 \( 1 + 5.07e23T + 1.35e48T^{2} \)
73 \( 1 + (-9.82e23 + 9.82e23i)T - 2.79e48iT^{2} \)
79 \( 1 - 1.32e24iT - 2.17e49T^{2} \)
83 \( 1 + (-5.17e24 + 5.17e24i)T - 7.87e49iT^{2} \)
89 \( 1 + 1.90e25iT - 4.83e50T^{2} \)
97 \( 1 + (-2.71e25 - 2.71e25i)T + 4.52e51iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.06734020508147055145728837923, −16.71395024791864433353117548651, −14.83279570065015662845267299871, −12.07812685698749052229670186283, −10.71613708729772315863058146175, −10.21151579639973839052182263667, −8.231487072541769728269680680821, −5.64423086167036718030899869265, −3.30162321290456950324212938371, −1.62189171419536079318667675150, 0.41066159851914254122066019648, 1.15546791509885059281878589080, 5.15118457256229721245297719452, 6.66262509689621471469839768232, 7.926732222318755409140204874321, 9.453669546292637651820295030123, 11.62669608276078455044110305265, 13.45352143992569805378665899424, 15.69168249832416882770133875715, 17.08378327604692731316186617058

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