Properties

Label 2-5-5.3-c16-0-6
Degree $2$
Conductor $5$
Sign $-0.906 - 0.423i$
Analytic cond. $8.11622$
Root an. cond. $2.84889$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (233. − 233. i)2-s + (−4.70e3 − 4.70e3i)3-s − 4.31e4i·4-s + (−3.87e5 + 4.54e4i)5-s − 2.19e6·6-s + (−2.16e5 + 2.16e5i)7-s + (5.20e6 + 5.20e6i)8-s + 1.28e6i·9-s + (−7.98e7 + 1.01e8i)10-s − 2.53e8·11-s + (−2.03e8 + 2.03e8i)12-s + (−7.01e8 − 7.01e8i)13-s + 1.00e8i·14-s + (2.04e9 + 1.61e9i)15-s + 5.26e9·16-s + (1.16e9 − 1.16e9i)17-s + ⋯
L(s)  = 1  + (0.910 − 0.910i)2-s + (−0.717 − 0.717i)3-s − 0.659i·4-s + (−0.993 + 0.116i)5-s − 1.30·6-s + (−0.0375 + 0.0375i)7-s + (0.310 + 0.310i)8-s + 0.0298i·9-s + (−0.798 + 1.01i)10-s − 1.18·11-s + (−0.472 + 0.472i)12-s + (−0.860 − 0.860i)13-s + 0.0683i·14-s + (0.796 + 0.629i)15-s + 1.22·16-s + (0.166 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.906 - 0.423i$
Analytic conductor: \(8.11622\)
Root analytic conductor: \(2.84889\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :8),\ -0.906 - 0.423i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.209376 + 0.942989i\)
\(L(\frac12)\) \(\approx\) \(0.209376 + 0.942989i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.87e5 - 4.54e4i)T \)
good2 \( 1 + (-233. + 233. i)T - 6.55e4iT^{2} \)
3 \( 1 + (4.70e3 + 4.70e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (2.16e5 - 2.16e5i)T - 3.32e13iT^{2} \)
11 \( 1 + 2.53e8T + 4.59e16T^{2} \)
13 \( 1 + (7.01e8 + 7.01e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-1.16e9 + 1.16e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 3.15e10iT - 2.88e20T^{2} \)
23 \( 1 + (-2.72e10 - 2.72e10i)T + 6.13e21iT^{2} \)
29 \( 1 + 2.93e11iT - 2.50e23T^{2} \)
31 \( 1 + 2.25e11T + 7.27e23T^{2} \)
37 \( 1 + (4.39e12 - 4.39e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 1.29e13T + 6.37e25T^{2} \)
43 \( 1 + (1.86e12 + 1.86e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (-1.81e13 + 1.81e13i)T - 5.66e26iT^{2} \)
53 \( 1 + (1.95e13 + 1.95e13i)T + 3.87e27iT^{2} \)
59 \( 1 + 1.51e14iT - 2.15e28T^{2} \)
61 \( 1 + 6.79e13T + 3.67e28T^{2} \)
67 \( 1 + (-6.76e13 + 6.76e13i)T - 1.64e29iT^{2} \)
71 \( 1 + 2.56e14T + 4.16e29T^{2} \)
73 \( 1 + (7.68e14 + 7.68e14i)T + 6.50e29iT^{2} \)
79 \( 1 + 1.48e15iT - 2.30e30T^{2} \)
83 \( 1 + (-1.97e15 - 1.97e15i)T + 5.07e30iT^{2} \)
89 \( 1 + 2.36e15iT - 1.54e31T^{2} \)
97 \( 1 + (3.97e15 - 3.97e15i)T - 6.14e31iT^{2} \)
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

−19.19675996209597287528881563746, −17.53083233150028423018793710384, −15.33639211178947480195518493741, −13.14356162477693823710301616812, −12.12450702252496307383710284477, −10.93650895372630602805423842284, −7.49670541338625454604816538948, −5.02198372615212692129361508528, −2.88954408001752653467619848818, −0.39641366595429618069398260963, 4.17469482125694636681492872773, 5.43411015184544664196208611284, 7.54858609782699029410654367019, 10.52070156875966359097470458471, 12.44106769493788917224181862541, 14.50966421653476819478244695435, 15.91215657933600922697864974132, 16.60750670246183011707753426761, 19.09998697542673814602583410005, 21.22187302783207582381876570026

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