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Properties

Label 1-15e2-225.13-r1-0-0
Degree $1$
Conductor $225$
Sign $-0.988 - 0.152i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
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  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.866 + 0.5i)7-s + (0.587 − 0.809i)8-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)13-s + (−0.669 + 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (0.406 − 0.913i)22-s + (0.743 + 0.669i)23-s + 26-s + (−0.587 − 0.809i)28-s + (0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.866 + 0.5i)7-s + (0.587 − 0.809i)8-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)13-s + (−0.669 + 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (0.406 − 0.913i)22-s + (0.743 + 0.669i)23-s + 26-s + (−0.587 − 0.809i)28-s + (0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.988 - 0.152i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ -0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06050364918 + 0.7863150956i\)
\(L(\frac12)\) \(\approx\) \(-0.06050364918 + 0.7863150956i\)
\(L(1)\) \(\approx\) \(0.6609738055 + 0.4531909239i\)
\(L(1)\) \(\approx\) \(0.6609738055 + 0.4531909239i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.207 - 0.978i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.994 + 0.104i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (0.994 + 0.104i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−26.31674864642083883979843135483, −24.65942945520193938893025179672, −23.72571848562931994796838552213, −22.80828566972396184226726873477, −21.78656362845176560751759025643, −20.78800189133764112661603259227, −20.38585252718672922235190119995, −19.12350156486775845137139139827, −18.27422613827829770213985948848, −17.4817154577675097090081836645, −16.47505382268843972539168948792, −15.07046215246216906582024479964, −13.843651398292091284907587807096, −13.278110167618158113028508019, −11.859847737339432917678564363746, −11.23379962032569001307606543379, −10.20762549441236124302287025446, −9.18727086898092938240391959355, −8.07856643919182795800084675601, −7.07064487981690179451648520523, −5.06371946917565700539950555254, −4.42664759916277107805187634707, −2.88189157101808252453923252463, −1.76635487761246606573810504145, −0.28526190874020454337640586562, 1.4896657213936508110055274773, 3.33271455727855886833798561123, 5.06320419888612947596120678237, 5.465931547246889148883563042911, 6.95304473571222865458434605805, 8.04871327180397857519829218527, 8.6383347751338828643417097681, 10.03079869153883304079174187668, 10.96164260660073962136494146220, 12.4791845458027649830493170359, 13.43132354281365588020413311766, 14.553006296613856700755096757263, 15.32574550760415236796554356694, 16.10142589436031404391693247760, 17.40560546817118604595781053098, 17.98133223095553999216998859566, 18.84064258581969563021735286876, 20.05713625974063823473262944025, 21.277886475282693557442906782957, 22.15597171993130048901209348852, 23.25541148934482390782141057125, 24.0386178647504333678726531670, 24.85305467810759930724592334069, 25.62239315579484253486623330381, 26.74114045217796131172948638106

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