Properties

Label 2-35e2-5.4-c1-0-50
Degree $2$
Conductor $1225$
Sign $-0.447 + 0.894i$
Analytic cond. $9.78167$
Root an. cond. $3.12756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.41i·3-s + 3.41·6-s + 2.82i·8-s − 2.82·9-s − 5.82·11-s − 1.58i·13-s − 4.00·16-s − 5.24i·17-s − 4i·18-s − 6·19-s − 8.24i·22-s − 4.58i·23-s + 6.82·24-s + 2.24·26-s − 0.414i·27-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.39i·3-s + 1.39·6-s + 0.999i·8-s − 0.942·9-s − 1.75·11-s − 0.439i·13-s − 1.00·16-s − 1.27i·17-s − 0.942i·18-s − 1.37·19-s − 1.75i·22-s − 0.956i·23-s + 1.39·24-s + 0.439·26-s − 0.0797i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(9.78167\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8110574785\)
\(L(\frac12)\) \(\approx\) \(0.8110574785\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 - 1.41iT - 2T^{2} \)
3 \( 1 + 2.41iT - 3T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 + 1.58iT - 13T^{2} \)
17 \( 1 + 5.24iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 6.24iT - 37T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 1.24iT - 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 - 0.242iT - 67T^{2} \)
71 \( 1 + 8.82T + 71T^{2} \)
73 \( 1 + 8.48iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 4.75iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.048932867088024778413584334170, −8.125257312603343690969921542023, −7.72207077803613579800226696945, −7.07473151241995711956821155016, −6.30625124040644672737329991808, −5.56780775405479700421169115324, −4.69419364329223309662316416713, −2.72720781824124058285194022912, −2.17246388166365842908076329541, −0.30563185975764321630357699312, 1.87015858748459871769252328118, 2.91641999633163643795672602249, 3.81586321905988346677616454748, 4.54104191858632295215648241150, 5.51870060720393095539463448609, 6.53326402621313878751977596704, 7.76037226018954255244563242236, 8.620502204500159371870507338302, 9.601291724373722324816219953672, 10.16267491440172677444891932462

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