Properties

Label 2-1875-5.4-c1-0-35
Degree $2$
Conductor $1875$
Sign $-i$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
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  − 0.0898i·2-s + i·3-s + 1.99·4-s + 0.0898·6-s + 4.36i·7-s − 0.358i·8-s − 9-s + 4.39·11-s + 1.99i·12-s + 1.98i·13-s + 0.391·14-s + 3.95·16-s + 0.997i·17-s + 0.0898i·18-s − 1.35·19-s + ⋯
L(s)  = 1  − 0.0635i·2-s + 0.577i·3-s + 0.995·4-s + 0.0366·6-s + 1.64i·7-s − 0.126i·8-s − 0.333·9-s + 1.32·11-s + 0.575i·12-s + 0.549i·13-s + 0.104·14-s + 0.987·16-s + 0.241i·17-s + 0.0211i·18-s − 0.309·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-i$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.360464941\)
\(L(\frac12)\) \(\approx\) \(2.360464941\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + 0.0898iT - 2T^{2} \)
7 \( 1 - 4.36iT - 7T^{2} \)
11 \( 1 - 4.39T + 11T^{2} \)
13 \( 1 - 1.98iT - 13T^{2} \)
17 \( 1 - 0.997iT - 17T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
23 \( 1 + 2.35iT - 23T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 + 3.67T + 31T^{2} \)
37 \( 1 + 1.43iT - 37T^{2} \)
41 \( 1 + 5.98T + 41T^{2} \)
43 \( 1 + 2.68iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + 6.68T + 59T^{2} \)
61 \( 1 + 9.45T + 61T^{2} \)
67 \( 1 - 12.9iT - 67T^{2} \)
71 \( 1 - 7.32T + 71T^{2} \)
73 \( 1 - 0.424iT - 73T^{2} \)
79 \( 1 - 6.35T + 79T^{2} \)
83 \( 1 + 0.737iT - 83T^{2} \)
89 \( 1 - 9.78T + 89T^{2} \)
97 \( 1 - 0.0337iT - 97T^{2} \)
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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

−9.314923885700460834007407465494, −8.793691442116926391393080759781, −8.039734498971952547897688852890, −6.72950592920919142799440558586, −6.32853420077471931877083677456, −5.52406029160529845990673252021, −4.50490089076100458497674586668, −3.42938706387538025976643467449, −2.52552978681891474943384107472, −1.62552315832980889346852496148, 0.887606847308669710052280881025, 1.72733939672791954151113898787, 3.11337912834457163866977951110, 3.86171754776406903080667511288, 4.98107650752665993856903368305, 6.29328976244314287339313678528, 6.63111402360947756535292676348, 7.42016977323970355096749334964, 7.909183304195615313312195937649, 8.955050549820015677095823994961

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