↑

Properties

Label 2-5-5.2-c26-0-1
Degree $2$
Conductor $5$
Sign $0.914 + 0.404i$
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 + (−8.43e4 + 8.43e4i)3-s + 9.35e7i·4-s + (−6.90e8 − 1.00e9i)5-s + 1.51e9·6-s + (−8.97e10 − 8.97e10i)7-s + (2.37e11 − 2.37e11i)8-s + 2.52e12i·9-s + (−2.83e12 + 1.52e13i)10-s − 9.83e12·11-s + (−7.89e12 − 7.89e12i)12-s + (−2.69e14 + 2.69e14i)13-s + 1.60e15i·14-s + (1.43e14 + 2.66e13i)15-s + 2.02e15·16-s + (−8.05e15 − 8.05e15i)17-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s + (−0.0529 + 0.0529i)3-s + 1.39i·4-s + (−0.565 − 0.824i)5-s + 0.115·6-s + (−0.925 − 0.925i)7-s + (0.431 − 0.431i)8-s + 0.994i·9-s + (−0.283 + 1.52i)10-s − 0.284·11-s + (−0.0737 − 0.0737i)12-s + (−0.889 + 0.889i)13-s + 2.02i·14-s + (0.0735 + 0.0136i)15-s + 0.449·16-s + (−0.813 − 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.914 + 0.404i$
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: \((2,\ 5,\ (\ :13),\ 0.914 + 0.404i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.336160 - 0.0709370i\)
\(L(\frac12)\) \(\approx\) \(0.336160 - 0.0709370i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (6.90e8 + 1.00e9i)T \)
good2 \( 1 + (8.96e3 + 8.96e3i)T + 6.71e7iT^{2} \)
3 \( 1 + (8.43e4 - 8.43e4i)T - 2.54e12iT^{2} \)
7 \( 1 + (8.97e10 + 8.97e10i)T + 9.38e21iT^{2} \)
11 \( 1 + 9.83e12T + 1.19e27T^{2} \)
13 \( 1 + (2.69e14 - 2.69e14i)T - 9.17e28iT^{2} \)
17 \( 1 + (8.05e15 + 8.05e15i)T + 9.81e31iT^{2} \)
19 \( 1 + 1.22e15iT - 1.76e33T^{2} \)
23 \( 1 + (-2.54e17 + 2.54e17i)T - 2.54e35iT^{2} \)
29 \( 1 + 9.37e18iT - 1.05e38T^{2} \)
31 \( 1 - 3.37e19T + 5.96e38T^{2} \)
37 \( 1 + (1.40e20 + 1.40e20i)T + 5.93e40iT^{2} \)
41 \( 1 - 5.50e20T + 8.55e41T^{2} \)
43 \( 1 + (2.83e20 - 2.83e20i)T - 2.95e42iT^{2} \)
47 \( 1 + (-6.66e21 - 6.66e21i)T + 2.98e43iT^{2} \)
53 \( 1 + (-3.12e22 + 3.12e22i)T - 6.77e44iT^{2} \)
59 \( 1 - 2.03e23iT - 1.10e46T^{2} \)
61 \( 1 + 1.20e23T + 2.62e46T^{2} \)
67 \( 1 + (3.15e23 + 3.15e23i)T + 3.00e47iT^{2} \)
71 \( 1 + 1.27e24T + 1.35e48T^{2} \)
73 \( 1 + (9.38e23 - 9.38e23i)T - 2.79e48iT^{2} \)
79 \( 1 + 2.70e24iT - 2.17e49T^{2} \)
83 \( 1 + (-3.59e24 + 3.59e24i)T - 7.87e49iT^{2} \)
89 \( 1 - 1.92e25iT - 4.83e50T^{2} \)
97 \( 1 + (-7.59e25 - 7.59e25i)T + 4.52e51iT^{2} \)
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

−17.17050657510482532714506331838, −16.20182627068734833387976171374, −13.34659138921824879822949124893, −11.83044847574208139900021285990, −10.39528760157500719612804606881, −9.095329736621737976546451471606, −7.48501900357721465014659048976, −4.43125290561788459283407253153, −2.48578735671856114554324137789, −0.71542187238483176314632032625, 0.28492845906432360597830009492, 3.05958852527906658458772531454, 6.02670829344873786109419706521, 7.13824295242285300183436967084, 8.739950574347095929191067312434, 10.15207483121884088587020193162, 12.34799516527364469494465088415, 15.11165422727027373838696098015, 15.55196942623684376223480942519, 17.41205203156660078320785168880

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