Properties

Label 2-5-5.4-c33-0-5
Degree $2$
Conductor $5$
Sign $-0.997 - 0.0764i$
Analytic cond. $34.4914$
Root an. cond. $5.87293$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.04e4i·2-s + 1.28e8i·3-s + 6.95e9·4-s + (3.40e11 + 2.60e10i)5-s − 5.19e12·6-s − 4.50e13i·7-s + 6.28e14i·8-s − 1.09e16·9-s + (−1.05e15 + 1.37e16i)10-s + 7.97e16·11-s + 8.94e17i·12-s + 2.76e18i·13-s + 1.81e18·14-s + (−3.35e18 + 4.37e19i)15-s + 3.43e19·16-s − 5.61e19i·17-s + ⋯
L(s)  = 1  + 0.435i·2-s + 1.72i·3-s + 0.809·4-s + (0.997 + 0.0764i)5-s − 0.751·6-s − 0.512i·7-s + 0.789i·8-s − 1.97·9-s + (−0.0333 + 0.434i)10-s + 0.523·11-s + 1.39i·12-s + 1.15i·13-s + 0.223·14-s + (−0.131 + 1.71i)15-s + 0.465·16-s − 0.279i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0764i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.997 - 0.0764i$
Analytic conductor: \(34.4914\)
Root analytic conductor: \(5.87293\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :33/2),\ -0.997 - 0.0764i)\)

Particular Values

\(L(17)\) \(\approx\) \(2.727027740\)
\(L(\frac12)\) \(\approx\) \(2.727027740\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.40e11 - 2.60e10i)T \)
good2 \( 1 - 4.04e4iT - 8.58e9T^{2} \)
3 \( 1 - 1.28e8iT - 5.55e15T^{2} \)
7 \( 1 + 4.50e13iT - 7.73e27T^{2} \)
11 \( 1 - 7.97e16T + 2.32e34T^{2} \)
13 \( 1 - 2.76e18iT - 5.75e36T^{2} \)
17 \( 1 + 5.61e19iT - 4.02e40T^{2} \)
19 \( 1 + 1.90e21T + 1.58e42T^{2} \)
23 \( 1 - 4.97e22iT - 8.65e44T^{2} \)
29 \( 1 - 1.05e24T + 1.81e48T^{2} \)
31 \( 1 + 5.13e24T + 1.64e49T^{2} \)
37 \( 1 + 1.64e25iT - 5.63e51T^{2} \)
41 \( 1 + 3.83e26T + 1.66e53T^{2} \)
43 \( 1 - 6.26e25iT - 8.02e53T^{2} \)
47 \( 1 + 5.32e27iT - 1.51e55T^{2} \)
53 \( 1 + 8.17e27iT - 7.96e56T^{2} \)
59 \( 1 - 2.46e29T + 2.74e58T^{2} \)
61 \( 1 + 5.26e28T + 8.23e58T^{2} \)
67 \( 1 + 2.38e30iT - 1.82e60T^{2} \)
71 \( 1 - 2.98e30T + 1.23e61T^{2} \)
73 \( 1 - 8.89e30iT - 3.08e61T^{2} \)
79 \( 1 - 2.50e31T + 4.18e62T^{2} \)
83 \( 1 + 2.29e31iT - 2.13e63T^{2} \)
89 \( 1 + 6.67e31T + 2.13e64T^{2} \)
97 \( 1 - 6.85e32iT - 3.65e65T^{2} \)
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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

−16.51727736855212842569027789654, −15.14678228729638862506743559125, −14.07554188043248718570606900623, −11.34607918386518065239427090259, −10.22543639008205003105370843288, −8.993490717815409661240755737254, −6.65719925957877521660650061533, −5.26441652291340084919914627992, −3.74073509570432126438788564758, −1.99859078826092672289999521555, 0.75660764690268216491095841024, 1.90115947432172712045360063780, 2.67214706485924035350386726647, 5.94463569463661448418083900354, 6.75862327556656403299195551437, 8.453915973014617028239977377399, 10.63060231239519384347198888410, 12.32393539022460693613928443233, 12.96671471089935560076614503743, 14.67665416700703084532413933277

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