Properties

Label 2-1425-5.4-c1-0-36
Degree $2$
Conductor $1425$
Sign $0.894 + 0.447i$
Analytic cond. $11.3786$
Root an. cond. $3.37323$
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  + i·2-s i·3-s + 4-s + 6-s + 3i·8-s − 9-s i·12-s − 6i·13-s − 16-s − 6i·17-s i·18-s + 19-s − 4i·23-s + 3·24-s + 6·26-s + i·27-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s + 0.5·4-s + 0.408·6-s + 1.06i·8-s − 0.333·9-s − 0.288i·12-s − 1.66i·13-s − 0.250·16-s − 1.45i·17-s − 0.235i·18-s + 0.229·19-s − 0.834i·23-s + 0.612·24-s + 1.17·26-s + 0.192i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(11.3786\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.902484376\)
\(L(\frac12)\) \(\approx\) \(1.902484376\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
19 \( 1 - T \)
good2 \( 1 - iT - 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.306923894793833202916130817722, −8.361041388099766347814370613467, −7.67119098712522108772256183766, −7.19754419177287008334418952274, −6.22089904611708490906937569147, −5.60472308218664705087760240801, −4.72874787250293413222882288577, −3.07677877940233359010097148313, −2.42548055461719169895524582371, −0.789059154885745076802509674053, 1.44182792165924474306904393675, 2.39872836178547490586255406985, 3.63453873133670496063262578144, 4.15289818673009171271111122982, 5.36771957069121912469523980965, 6.44638916603075771355916384470, 6.97258596355170110189378655983, 8.196864413088494728343702987135, 8.948195871907324000489520010569, 10.00176351171452592205052679002

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