Properties

Label 2-5-5.4-c23-0-8
Degree $2$
Conductor $5$
Sign $-0.897 - 0.441i$
Analytic cond. $16.7602$
Root an. cond. $4.09392$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.67e3i·2-s − 3.30e5i·3-s + 5.56e6·4-s + (−9.79e7 − 4.82e7i)5-s − 5.55e8·6-s − 4.39e9i·7-s − 2.34e10i·8-s − 1.51e10·9-s + (−8.10e10 + 1.64e11i)10-s − 6.96e10·11-s − 1.84e12i·12-s − 1.33e12i·13-s − 7.38e12·14-s + (−1.59e13 + 3.23e13i)15-s + 7.31e12·16-s + 1.24e14i·17-s + ⋯
L(s)  = 1  − 0.579i·2-s − 1.07i·3-s + 0.663·4-s + (−0.897 − 0.441i)5-s − 0.624·6-s − 0.840i·7-s − 0.964i·8-s − 0.161·9-s + (−0.256 + 0.520i)10-s − 0.0736·11-s − 0.715i·12-s − 0.206i·13-s − 0.487·14-s + (−0.475 + 0.966i)15-s + 0.103·16-s + 0.884i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.897 - 0.441i$
Analytic conductor: \(16.7602\)
Root analytic conductor: \(4.09392\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :23/2),\ -0.897 - 0.441i)\)

Particular Values

\(L(12)\) \(\approx\) \(0.331701 + 1.42463i\)
\(L(\frac12)\) \(\approx\) \(0.331701 + 1.42463i\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (9.79e7 + 4.82e7i)T \)
good2 \( 1 + 1.67e3iT - 8.38e6T^{2} \)
3 \( 1 + 3.30e5iT - 9.41e10T^{2} \)
7 \( 1 + 4.39e9iT - 2.73e19T^{2} \)
11 \( 1 + 6.96e10T + 8.95e23T^{2} \)
13 \( 1 + 1.33e12iT - 4.17e25T^{2} \)
17 \( 1 - 1.24e14iT - 1.99e28T^{2} \)
19 \( 1 + 8.63e14T + 2.57e29T^{2} \)
23 \( 1 - 5.41e15iT - 2.08e31T^{2} \)
29 \( 1 - 1.26e16T + 4.31e33T^{2} \)
31 \( 1 - 1.60e17T + 2.00e34T^{2} \)
37 \( 1 + 1.33e18iT - 1.17e36T^{2} \)
41 \( 1 + 4.85e18T + 1.24e37T^{2} \)
43 \( 1 + 8.33e18iT - 3.71e37T^{2} \)
47 \( 1 - 1.44e19iT - 2.87e38T^{2} \)
53 \( 1 + 1.04e20iT - 4.55e39T^{2} \)
59 \( 1 - 1.12e20T + 5.36e40T^{2} \)
61 \( 1 + 2.79e20T + 1.15e41T^{2} \)
67 \( 1 + 1.54e21iT - 9.99e41T^{2} \)
71 \( 1 - 1.52e21T + 3.79e42T^{2} \)
73 \( 1 - 1.16e20iT - 7.18e42T^{2} \)
79 \( 1 - 2.27e21T + 4.42e43T^{2} \)
83 \( 1 - 8.52e21iT - 1.37e44T^{2} \)
89 \( 1 - 4.39e22T + 6.85e44T^{2} \)
97 \( 1 + 9.63e22iT - 4.96e45T^{2} \)
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

−17.10754837025697976266569992321, −15.45281179136853472645597852923, −13.14330200971489759853485380586, −12.12144953991839731210385464714, −10.63912462027189993352476778201, −7.944875546233673634268498939593, −6.71880660424365020539255007267, −3.85002650887680936637508637972, −1.80719543815650555416460328598, −0.54918841304476118423356014456, 2.72124442008284013739821915990, 4.62282566720187704374397931288, 6.60271012839081675363604789984, 8.442963856258696456721143919883, 10.53437726610447988833099878690, 11.90093927743091530866810949859, 14.85964844268394456108680382549, 15.52596172016776571965777648848, 16.61002813807208657050406835133, 18.81981264981750345039632904876

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