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Properties

Label 1-35e2-1225.464-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.524 - 0.851i$
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  + (0.447 − 0.894i)2-s + (0.999 + 0.0299i)3-s + (−0.599 − 0.800i)4-s + (0.473 − 0.880i)6-s + (−0.983 + 0.178i)8-s + (0.998 + 0.0598i)9-s + (0.998 − 0.0598i)11-s + (−0.575 − 0.817i)12-s + (−0.936 − 0.351i)13-s + (−0.280 + 0.959i)16-s + (−0.992 − 0.119i)17-s + (0.5 − 0.866i)18-s + (0.913 + 0.406i)19-s + (0.393 − 0.919i)22-s + (−0.420 − 0.907i)23-s + (−0.988 + 0.149i)24-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)2-s + (0.999 + 0.0299i)3-s + (−0.599 − 0.800i)4-s + (0.473 − 0.880i)6-s + (−0.983 + 0.178i)8-s + (0.998 + 0.0598i)9-s + (0.998 − 0.0598i)11-s + (−0.575 − 0.817i)12-s + (−0.936 − 0.351i)13-s + (−0.280 + 0.959i)16-s + (−0.992 − 0.119i)17-s + (0.5 − 0.866i)18-s + (0.913 + 0.406i)19-s + (0.393 − 0.919i)22-s + (−0.420 − 0.907i)23-s + (−0.988 + 0.149i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.234025159 - 2.208766938i\)
\(L(\frac12)\) \(\approx\) \(1.234025159 - 2.208766938i\)
\(L(1)\) \(\approx\) \(1.385738073 - 1.006433004i\)
\(L(1)\) \(\approx\) \(1.385738073 - 1.006433004i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.447 - 0.894i)T \)
3 \( 1 + (0.999 + 0.0299i)T \)
11 \( 1 + (0.998 - 0.0598i)T \)
13 \( 1 + (-0.936 - 0.351i)T \)
17 \( 1 + (-0.992 - 0.119i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.420 - 0.907i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.575 - 0.817i)T \)
41 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.712 - 0.701i)T \)
53 \( 1 + (0.599 + 0.800i)T \)
59 \( 1 + (0.971 - 0.237i)T \)
61 \( 1 + (0.0149 - 0.999i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (0.163 - 0.986i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (-0.772 - 0.635i)T \)
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

−21.69530837162361064112570412680, −20.67747507086960872678348299105, −19.67059868944277084523718142254, −19.40590511902554179734572037875, −17.98033670381830156844128511821, −17.69542514677715537468447959695, −16.54141857640774303812132303605, −15.87764331250196819442746237554, −15.11045106170473461284963340571, −14.41252781438990433779568850553, −13.89433256012694843146941959308, −13.14019085466183232911222849291, −12.25884392116248280253439102117, −11.53142929358893785889385789050, −10.009441758209048157660906034688, −9.26371013934659674550327354763, −8.71896062551370731120492561593, −7.75210178962127012076783329933, −7.009689873091692582824410967894, −6.4466955722950136731387091162, −5.07040536247105556201813148801, −4.40641000394004865227894089606, −3.49573607259068890182662924568, −2.68571529204465427745824198725, −1.42503114593749552956352093409, 0.79183278952955646235344524157, 2.07256701160867044995299134128, 2.57516095733142435636051752405, 3.73044476217167561886421990658, 4.24236908904531511430958999745, 5.248728154359426367197601620320, 6.40046764572544158122543313486, 7.35519740658699564936505578830, 8.44829976250672316130952557559, 9.18030484699850817640248808311, 9.84383603165189741656158427186, 10.57009789215382462097704958632, 11.68693312295408723849271556675, 12.31823628231014084903782940322, 13.11788912963969504858989141026, 14.003797333324063783723184920232, 14.38458674401447181665847541455, 15.20707567437154706736903991167, 15.97594962111770352984311994775, 17.25329603151795093617638429004, 18.047529397840113988164440006789, 18.93645145414121751828637432708, 19.542001062176084375529824427192, 20.16459603537630368270297784015, 20.66078473391308189774762451407

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