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Properties

Label 1-35e2-1225.8-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.940 + 0.340i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (−0.919 + 0.393i)2-s + (0.834 − 0.550i)3-s + (0.691 − 0.722i)4-s + (−0.550 + 0.834i)6-s + (−0.351 + 0.936i)8-s + (0.393 − 0.919i)9-s + (−0.393 − 0.919i)11-s + (0.178 − 0.983i)12-s + (0.657 + 0.753i)13-s + (−0.0448 − 0.998i)16-s + (−0.722 + 0.691i)17-s + i·18-s + (−0.309 − 0.951i)19-s + (0.722 + 0.691i)22-s + (0.178 + 0.983i)23-s + (0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (−0.919 + 0.393i)2-s + (0.834 − 0.550i)3-s + (0.691 − 0.722i)4-s + (−0.550 + 0.834i)6-s + (−0.351 + 0.936i)8-s + (0.393 − 0.919i)9-s + (−0.393 − 0.919i)11-s + (0.178 − 0.983i)12-s + (0.657 + 0.753i)13-s + (−0.0448 − 0.998i)16-s + (−0.722 + 0.691i)17-s + i·18-s + (−0.309 − 0.951i)19-s + (0.722 + 0.691i)22-s + (0.178 + 0.983i)23-s + (0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.940 + 0.340i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.940 + 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02295542016 - 0.1307858727i\)
\(L(\frac12)\) \(\approx\) \(0.02295542016 - 0.1307858727i\)
\(L(1)\) \(\approx\) \(0.7945827699 - 0.09793906030i\)
\(L(1)\) \(\approx\) \(0.7945827699 - 0.09793906030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.919 + 0.393i)T \)
3 \( 1 + (0.834 - 0.550i)T \)
11 \( 1 + (-0.393 - 0.919i)T \)
13 \( 1 + (0.657 + 0.753i)T \)
17 \( 1 + (-0.722 + 0.691i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (0.178 + 0.983i)T \)
29 \( 1 + (-0.134 - 0.990i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.178 - 0.983i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (-0.781 + 0.623i)T \)
47 \( 1 + (0.512 - 0.858i)T \)
53 \( 1 + (-0.722 - 0.691i)T \)
59 \( 1 + (0.0448 + 0.998i)T \)
61 \( 1 + (0.473 - 0.880i)T \)
67 \( 1 + (0.951 - 0.309i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (-0.657 + 0.753i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.512 + 0.858i)T \)
89 \( 1 + (-0.753 - 0.657i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−20.86022501307581231835018192238, −20.3932267738736658447844645220, −20.22389277093576977593798511933, −18.91281794401928830820467255113, −18.552737086466140282128569973734, −17.65400177803044011136853203269, −16.759652753126493183510254138472, −15.9777592394194850071318779277, −15.34841113730835507659445213303, −14.66640854049793384455534263511, −13.453545669923477574139988214, −12.83595857535552960503289792491, −11.90796138491186499958814754546, −10.77351930324982966167447893035, −10.335398369026974279959097089810, −9.565051777685364296239766082264, −8.73873456296482514911398506261, −8.10704245680538553090737437814, −7.35952676448514301971999586953, −6.38898206193510638413948180439, −4.99243385884487424659162989353, −4.05404752715696558499501463588, −3.10803011558072730623181617012, −2.3548748170778225541773609747, −1.42849990295918882581369242560, 0.030924652015700070229087619025, 1.17092868983384071132124815321, 2.0268612875426364519496585574, 2.954118569802499881374745874726, 4.02818435711238560469268761479, 5.431564442549533485003440064642, 6.45222692241743728125595661339, 6.92108159769731723982967550685, 8.00311768458667059735024859116, 8.57336369286162154576987216673, 9.13714810200594027962602033388, 10.05918436612380199214993162094, 11.11145245289377163855070133846, 11.63316867854184796042036720415, 12.98039095539006532579786363707, 13.6269223627860876009263882635, 14.34708427475435129582164191430, 15.38572599851714558704857794841, 15.7240260879928440176903191484, 16.78000715227602903893688881904, 17.61653263778569155932344923936, 18.271508460983805304812502138333, 19.07226230386148904596561741192, 19.4867807150280979814855568738, 20.19979523139818082107955107069

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