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Properties

Label 1-35e2-1225.134-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

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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: $\chi_{1225} (134, \cdot )$
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 \( 1 + (0.809 - 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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