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Properties

Label 1-35e2-1225.561-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.999 + 0.0282i$
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  + (−0.550 − 0.834i)2-s + (0.473 + 0.880i)3-s + (−0.393 + 0.919i)4-s + (0.473 − 0.880i)6-s + (0.983 − 0.178i)8-s + (−0.550 + 0.834i)9-s + (−0.550 − 0.834i)11-s + (−0.995 + 0.0896i)12-s + (0.936 + 0.351i)13-s + (−0.691 − 0.722i)16-s + (−0.393 − 0.919i)17-s + 18-s + (−0.809 + 0.587i)19-s + (−0.393 + 0.919i)22-s + (−0.995 − 0.0896i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (−0.550 − 0.834i)2-s + (0.473 + 0.880i)3-s + (−0.393 + 0.919i)4-s + (0.473 − 0.880i)6-s + (0.983 − 0.178i)8-s + (−0.550 + 0.834i)9-s + (−0.550 − 0.834i)11-s + (−0.995 + 0.0896i)12-s + (0.936 + 0.351i)13-s + (−0.691 − 0.722i)16-s + (−0.393 − 0.919i)17-s + 18-s + (−0.809 + 0.587i)19-s + (−0.393 + 0.919i)22-s + (−0.995 − 0.0896i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0282i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.999 + 0.0282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0001446066999 + 0.01025139903i\)
\(L(\frac12)\) \(\approx\) \(0.0001446066999 + 0.01025139903i\)
\(L(1)\) \(\approx\) \(0.6792550844 - 0.03411363204i\)
\(L(1)\) \(\approx\) \(0.6792550844 - 0.03411363204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.550 - 0.834i)T \)
3 \( 1 + (0.473 + 0.880i)T \)
11 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (-0.393 - 0.919i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.995 - 0.0896i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.995 + 0.0896i)T \)
41 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.963 + 0.266i)T \)
53 \( 1 + (-0.393 + 0.919i)T \)
59 \( 1 + (-0.691 - 0.722i)T \)
61 \( 1 + (0.858 + 0.512i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (0.936 - 0.351i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.963 - 0.266i)T \)
89 \( 1 + (0.936 - 0.351i)T \)
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

−20.368011000087937196315248379295, −19.83123061609637069889779736989, −19.12832872090166798741154076750, −18.12293556025383265570786593034, −17.9616487957992460742100497354, −17.107406772164873106243845000373, −16.09845591269583330032328040946, −15.23338392335362039013287492495, −14.80215275087306988062946760414, −13.78337456609537591075337418042, −13.145424992342154743008715918, −12.47836991733242577012711564089, −11.20388482478301279021527414538, −10.38605163428188843197749462368, −9.4918998433838274962666540631, −8.41982526425338393397549857378, −8.23251962085641366065164128577, −7.15810748646298567759839150191, −6.51369662973582062023093179924, −5.774178745549643645331255086176, −4.66383989034938545155587706009, −3.54759184982026642735008929682, −2.14531356131146591070678408774, −1.48654412299517116484295124943, −0.0044024926666677412020801970, 1.64257235968324097026429548091, 2.61775809608610477020384394951, 3.44742976939616412570961867528, 4.15446215386990775599694191724, 5.08719855220245427431564663242, 6.2393654327565426879173070116, 7.570706329491951799985011373660, 8.505999990150387198376069153987, 8.77551541812910176261055685901, 9.84117769716955020541532568670, 10.48936679879959589942446494448, 11.14203932074127462358205297317, 11.87642930646743129278696959047, 13.01868467762240790728856715893, 13.78039886821099204209772785967, 14.30560237491068254396428514086, 15.75637468264237160595374084669, 16.05990526284282247922969222032, 16.85831543679383176101703169638, 17.83998488774357192461304750009, 18.64825535260404156148121937949, 19.252902046060844510139097474729, 20.10681030881994674667411445412, 20.78182317654754092166721843020, 21.285638842086359951703412605

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