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Properties

Label 1-35e2-1225.29-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.0909 + 0.995i$
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  + (0.0448 + 0.998i)2-s + (0.691 + 0.722i)3-s + (−0.995 + 0.0896i)4-s + (−0.691 + 0.722i)6-s + (−0.134 − 0.990i)8-s + (−0.0448 + 0.998i)9-s + (−0.0448 − 0.998i)11-s + (−0.753 − 0.657i)12-s + (0.963 + 0.266i)13-s + (0.983 − 0.178i)16-s + (0.995 + 0.0896i)17-s − 18-s + (0.309 − 0.951i)19-s + (0.995 − 0.0896i)22-s + (−0.753 + 0.657i)23-s + (0.623 − 0.781i)24-s + ⋯
L(s)  = 1  + (0.0448 + 0.998i)2-s + (0.691 + 0.722i)3-s + (−0.995 + 0.0896i)4-s + (−0.691 + 0.722i)6-s + (−0.134 − 0.990i)8-s + (−0.0448 + 0.998i)9-s + (−0.0448 − 0.998i)11-s + (−0.753 − 0.657i)12-s + (0.963 + 0.266i)13-s + (0.983 − 0.178i)16-s + (0.995 + 0.0896i)17-s − 18-s + (0.309 − 0.951i)19-s + (0.995 − 0.0896i)22-s + (−0.753 + 0.657i)23-s + (0.623 − 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0909 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0909 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.0909 + 0.995i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.0909 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.312978953 + 1.438306916i\)
\(L(\frac12)\) \(\approx\) \(1.312978953 + 1.438306916i\)
\(L(1)\) \(\approx\) \(1.065820503 + 0.8040179320i\)
\(L(1)\) \(\approx\) \(1.065820503 + 0.8040179320i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.0448 + 0.998i)T \)
3 \( 1 + (0.691 + 0.722i)T \)
11 \( 1 + (-0.0448 - 0.998i)T \)
13 \( 1 + (0.963 + 0.266i)T \)
17 \( 1 + (0.995 + 0.0896i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.753 + 0.657i)T \)
29 \( 1 + (0.858 - 0.512i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.753 - 0.657i)T \)
41 \( 1 + (0.473 - 0.880i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.550 - 0.834i)T \)
53 \( 1 + (0.995 - 0.0896i)T \)
59 \( 1 + (0.983 - 0.178i)T \)
61 \( 1 + (-0.393 + 0.919i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (0.963 - 0.266i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.550 + 0.834i)T \)
89 \( 1 + (-0.963 + 0.266i)T \)
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

−20.68176113952232053139109894820, −20.2972370934994170357994639944, −19.442996187754982068389302293807, −18.73044514951925785373561814018, −18.09005743588332947783159269060, −17.59697489246674365169199209895, −16.36810829371128538917526302757, −15.2909121510872468134830467626, −14.29735182277571858754304107394, −14.00360023355718094563087728293, −12.99247889988763523054551404903, −12.27147994660084511562625390217, −11.95631312589642053331674766237, −10.58866122657528935610899770731, −10.01764167361937504739383585964, −9.11140906877455977611761215669, −8.28184047913809473492556395570, −7.68596766798352628625798404893, −6.50421258610552037517438443460, −5.539092720306839868670221562500, −4.37413670270332700737941835205, −3.49569268850831630923067197881, −2.76008389675290729956972598038, −1.708799767492089706998123261330, −1.04513272324587201521050894554, 0.94233462848617518774838819530, 2.60700879499115755949561337648, 3.680383364786177096775138491771, 4.136342135587401307945912402401, 5.40673796382624972316002289474, 5.865819217784047777281051343612, 7.07731758253434560950015537800, 7.9718746658344045594122283099, 8.58467032314642575649269987331, 9.273309127644360464047674769443, 10.09176484254333429560158492071, 10.976785725073567893045455977809, 12.033914088418331208772323615465, 13.39761752359850305437407503759, 13.70442893559649892610973462371, 14.39603421021930012607456807810, 15.33263400265940071358770443725, 15.94279368089828710809591583665, 16.401343780647806432324196166319, 17.31094910542137540513411020298, 18.213959877056798278134412487044, 19.10564995743677057183096442638, 19.563522461607750970147151192594, 20.91109395053199158740683676603, 21.294194241675314003405183802059

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