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Properties

Label 1-35e2-1225.729-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.340 + 0.940i$
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  + (0.393 − 0.919i)2-s + (0.550 − 0.834i)3-s + (−0.691 − 0.722i)4-s + (−0.550 − 0.834i)6-s + (−0.936 + 0.351i)8-s + (−0.393 − 0.919i)9-s + (−0.393 + 0.919i)11-s + (−0.983 + 0.178i)12-s + (−0.753 − 0.657i)13-s + (−0.0448 + 0.998i)16-s + (0.691 − 0.722i)17-s − 18-s + (0.309 − 0.951i)19-s + (0.691 + 0.722i)22-s + (−0.983 − 0.178i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (0.393 − 0.919i)2-s + (0.550 − 0.834i)3-s + (−0.691 − 0.722i)4-s + (−0.550 − 0.834i)6-s + (−0.936 + 0.351i)8-s + (−0.393 − 0.919i)9-s + (−0.393 + 0.919i)11-s + (−0.983 + 0.178i)12-s + (−0.753 − 0.657i)13-s + (−0.0448 + 0.998i)16-s + (0.691 − 0.722i)17-s − 18-s + (0.309 − 0.951i)19-s + (0.691 + 0.722i)22-s + (−0.983 − 0.178i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5785468323 - 0.8248740118i\)
\(L(\frac12)\) \(\approx\) \(-0.5785468323 - 0.8248740118i\)
\(L(1)\) \(\approx\) \(0.6242295895 - 0.9045837729i\)
\(L(1)\) \(\approx\) \(0.6242295895 - 0.9045837729i\)

Euler product

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

−21.6821673747696561258304642736, −21.23665942822145381405960652484, −20.30350016549798421801422695151, −19.28840179254374277585928928675, −18.648096021749787348982647462005, −17.59095161997727818200149156993, −16.58496673177117550649697084451, −16.35971789072340590057495163193, −15.52423616513044196298767289316, −14.6280439410392885521495536977, −14.177063747836188877320996141066, −13.53378292276446634297376182797, −12.43933641138292992068205241584, −11.683537586213605596407772980046, −10.37448518333612762682071359590, −9.84510464008046023726003942403, −8.71615450306713593657637292774, −8.289991806861946237151680757067, −7.432352863615974799392971257997, −6.36432593571039871242750881513, −5.38624576420185398970830638978, −4.86446282845910040449135549642, −3.58766868671741936833109506759, −3.371186329477639520091685655258, −1.91069369634351553316808791571, 0.30717645426587711393198751645, 1.50826159766454118658617248871, 2.51080024221611166281912148668, 2.95198000994510325408640068005, 4.17539829626945841723579386610, 5.08487329699435639765315620453, 5.986525709987757618073315323878, 7.12654008307209455507019476249, 7.84183642279119011042575718036, 8.80212161286902041661053684436, 9.83168815555171160077069363647, 10.14724111316913185948531110186, 11.67484510995633495401379413923, 11.94232405677020995284545066274, 12.88666053997841845890558739378, 13.41879722373212271526310535553, 14.20599819146155254939955395117, 14.978488026917676681281256756199, 15.5893705763942123259020997603, 17.13980569640333888344591269401, 17.898142028901276263794316008673, 18.36844127445681653090451279027, 19.29759821248728326037729890469, 19.864457478941191641859237228356, 20.534060775016695736119226238

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