Properties

Label 2-35e2-5.4-c1-0-26
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  − 0.414i·2-s + 2.41i·3-s + 1.82·4-s + 0.999·6-s − 1.58i·8-s − 2.82·9-s + 4.82·11-s + 4.41i·12-s + 0.828i·13-s + 3·16-s + 0.828i·17-s + 1.17i·18-s − 2.82·19-s − 1.99i·22-s + 2.41i·23-s + 3.82·24-s + ⋯
L(s)  = 1  − 0.292i·2-s + 1.39i·3-s + 0.914·4-s + 0.408·6-s − 0.560i·8-s − 0.942·9-s + 1.45·11-s + 1.27i·12-s + 0.229i·13-s + 0.750·16-s + 0.200i·17-s + 0.276i·18-s − 0.648·19-s − 0.426i·22-s + 0.503i·23-s + 0.781·24-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

\(L(1)\) \(\approx\) \(2.259065206\)
\(L(\frac12)\) \(\approx\) \(2.259065206\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414iT - 2T^{2} \)
3 \( 1 - 2.41iT - 3T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 0.828iT - 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 2.41iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 + 6.41iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 6.82iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 12.4iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 4.82iT - 73T^{2} \)
79 \( 1 + 9.17T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 - 2.65T + 89T^{2} \)
97 \( 1 + 0.343iT - 97T^{2} \)
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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

−10.02795301950428654051445891586, −9.235013740897063121689138489669, −8.542094527111828576060692044555, −7.28512582570874284425665893701, −6.47354007191764255579222000822, −5.66406056987650846328711775385, −4.38661202099322942867100061638, −3.85228022950889969231191287551, −2.84767181314724703982367585873, −1.48218084465149714280254421557, 1.08481166666999243480452966491, 2.02290776710342070148380350202, 3.04614136733202821277725526607, 4.44055623516635101789118366176, 5.85987753033674573529102344848, 6.53303384377324438420651122397, 6.87359940672443188489464927404, 7.84254484413015139162649363278, 8.397503062013790745299861998613, 9.457012877230165935576419998528

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