↑

Properties

Label 1-35e2-1225.64-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.0679 + 0.997i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.691 − 0.722i)2-s + (0.393 − 0.919i)3-s + (−0.0448 − 0.998i)4-s + (−0.393 − 0.919i)6-s + (−0.753 − 0.657i)8-s + (−0.691 − 0.722i)9-s + (−0.691 + 0.722i)11-s + (−0.936 − 0.351i)12-s + (−0.134 + 0.990i)13-s + (−0.995 + 0.0896i)16-s + (0.0448 − 0.998i)17-s − 18-s + (−0.809 + 0.587i)19-s + (0.0448 + 0.998i)22-s + (−0.936 + 0.351i)23-s + (−0.900 + 0.433i)24-s + ⋯
L(s)  = 1  + (0.691 − 0.722i)2-s + (0.393 − 0.919i)3-s + (−0.0448 − 0.998i)4-s + (−0.393 − 0.919i)6-s + (−0.753 − 0.657i)8-s + (−0.691 − 0.722i)9-s + (−0.691 + 0.722i)11-s + (−0.936 − 0.351i)12-s + (−0.134 + 0.990i)13-s + (−0.995 + 0.0896i)16-s + (0.0448 − 0.998i)17-s − 18-s + (−0.809 + 0.587i)19-s + (0.0448 + 0.998i)22-s + (−0.936 + 0.351i)23-s + (−0.900 + 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0679 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.0679 + 0.997i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.0679 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3607448987 - 0.3861339833i\)
\(L(\frac12)\) \(\approx\) \(-0.3607448987 - 0.3861339833i\)
\(L(1)\) \(\approx\) \(0.7915851437 - 0.8122762343i\)
\(L(1)\) \(\approx\) \(0.7915851437 - 0.8122762343i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.691 - 0.722i)T \)
3 \( 1 + (0.393 - 0.919i)T \)
11 \( 1 + (-0.691 + 0.722i)T \)
13 \( 1 + (-0.134 + 0.990i)T \)
17 \( 1 + (0.0448 - 0.998i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.936 + 0.351i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.936 - 0.351i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.473 - 0.880i)T \)
53 \( 1 + (0.0448 + 0.998i)T \)
59 \( 1 + (-0.995 + 0.0896i)T \)
61 \( 1 + (-0.550 - 0.834i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.0448 - 0.998i)T \)
73 \( 1 + (-0.134 - 0.990i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.473 + 0.880i)T \)
89 \( 1 + (0.134 + 0.990i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.6655272486632413367271711968, −21.20525217171755521934103820545, −20.39104732246817656902471761434, −19.64139203190811372900575556790, −18.57275916852032854737621974106, −17.516357109993387908279534432260, −16.91812420788217503969350176164, −16.04269373342478526531392658980, −15.5529070019321005634753264525, −14.769470032654207813563757013911, −14.26371278393777414793400048622, −13.08260846333554169645889955024, −12.88500901354821518766244046294, −11.44920062311623014972237183703, −10.786892443456930067139460016256, −9.90339670599375870049614206209, −8.79015300966030186552636775737, −8.18681224109243186568582902477, −7.55147311810811303039323016633, −6.11797284011241517821381235451, −5.64350567513495211009394986336, −4.6868241087788918066275615994, −3.89101150313627471685089962280, −3.09775268670316465598309311661, −2.256117014909500744720396942839, 0.1356072419619991752533807158, 1.822273195928012432825138333097, 2.03576770928968592617275141393, 3.21668193896940080126325783339, 4.091871125670781620160732000338, 5.1393636562528209895075292113, 5.99372937035545005964210750739, 6.96557989746180534136328169670, 7.59100642462770233194471429983, 8.874489702452521292549358136519, 9.49796027733125535255530512596, 10.51474318508068002861460112264, 11.382049809395191197945557989539, 12.34626999965126323682170036541, 12.51936700716279242453597474248, 13.68768856311616513116524891567, 14.04089730952266524504630189392, 14.87872322003707002082108594311, 15.68098512671592072287319220187, 16.76056913972630114321928292331, 17.927742217431977554069078795, 18.4576393512528779969911924505, 19.10258746290480320396388074400, 19.902232467365047028979430966910, 20.54159770209465797283446365483

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