Properties

Label 2-1875-5.4-c1-0-34
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  + 2.01i·2-s i·3-s − 2.07·4-s + 2.01·6-s − 1.01i·7-s − 0.153i·8-s − 9-s + 4.75·11-s + 2.07i·12-s − 0.103i·13-s + 2.05·14-s − 3.84·16-s + 5.83i·17-s − 2.01i·18-s + 0.724·19-s + ⋯
L(s)  = 1  + 1.42i·2-s − 0.577i·3-s − 1.03·4-s + 0.824·6-s − 0.385i·7-s − 0.0541i·8-s − 0.333·9-s + 1.43·11-s + 0.599i·12-s − 0.0287i·13-s + 0.549·14-s − 0.960·16-s + 1.41i·17-s − 0.475i·18-s + 0.166·19-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

\(L(1)\) \(\approx\) \(1.842890705\)
\(L(\frac12)\) \(\approx\) \(1.842890705\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.01iT - 2T^{2} \)
7 \( 1 + 1.01iT - 7T^{2} \)
11 \( 1 - 4.75T + 11T^{2} \)
13 \( 1 + 0.103iT - 13T^{2} \)
17 \( 1 - 5.83iT - 17T^{2} \)
19 \( 1 - 0.724T + 19T^{2} \)
23 \( 1 + 9.07iT - 23T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 + 4.02iT - 37T^{2} \)
41 \( 1 - 7.20T + 41T^{2} \)
43 \( 1 - 8.62iT - 43T^{2} \)
47 \( 1 - 8.19iT - 47T^{2} \)
53 \( 1 + 4.36iT - 53T^{2} \)
59 \( 1 - 4.91T + 59T^{2} \)
61 \( 1 - 6.96T + 61T^{2} \)
67 \( 1 - 9.91iT - 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 8.63iT - 73T^{2} \)
79 \( 1 + 2.48T + 79T^{2} \)
83 \( 1 - 4.24iT - 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + 6.69iT - 97T^{2} \)
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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

−8.916539910662673409589611298691, −8.505469281759571599595877301670, −7.72075578510887228914345212233, −6.93554336835637961425906757689, −6.32889152350805434593946173555, −5.91263429356089837216210925218, −4.61787101274520211749069353041, −3.96018982576065090004806938599, −2.43687507801925806742011346676, −1.04847052097090395091230611644, 0.881588497108119330516263791023, 2.05263220309808373504520191116, 3.11359544236190108690834639806, 3.76666222073788711143197245217, 4.63613796280613574309516929655, 5.55038397622286526137680539178, 6.64446093347621904570515253352, 7.49056675285161233758336494893, 8.886583320985587325172351336881, 9.197590769871001325102381127331

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