Properties

Label 2-75-3.2-c8-0-15
Degree $2$
Conductor $75$
Sign $-i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17i·2-s − 81i·3-s − 33·4-s + 1.37e3·6-s + 3.79e3i·8-s − 6.56e3·9-s + 2.67e3i·12-s − 7.28e4·16-s − 2.11e4i·17-s − 1.11e5i·18-s + 2.03e5·19-s + 5.50e5i·23-s + 3.07e5·24-s + 5.31e5i·27-s + 1.83e6·31-s − 2.68e5i·32-s + ⋯
L(s)  = 1  + 1.06i·2-s i·3-s − 0.128·4-s + 1.06·6-s + 0.925i·8-s − 9-s + 0.128i·12-s − 1.11·16-s − 0.252i·17-s − 1.06i·18-s + 1.56·19-s + 1.96i·23-s + 0.925·24-s + i·27-s + 1.98·31-s − 0.256i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.38932 + 1.38932i\)
\(L(\frac12)\) \(\approx\) \(1.38932 + 1.38932i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81iT \)
5 \( 1 \)
good2 \( 1 - 17iT - 256T^{2} \)
7 \( 1 + 5.76e6T^{2} \)
11 \( 1 - 2.14e8T^{2} \)
13 \( 1 + 8.15e8T^{2} \)
17 \( 1 + 2.11e4iT - 6.97e9T^{2} \)
19 \( 1 - 2.03e5T + 1.69e10T^{2} \)
23 \( 1 - 5.50e5iT - 7.83e10T^{2} \)
29 \( 1 - 5.00e11T^{2} \)
31 \( 1 - 1.83e6T + 8.52e11T^{2} \)
37 \( 1 + 3.51e12T^{2} \)
41 \( 1 - 7.98e12T^{2} \)
43 \( 1 + 1.16e13T^{2} \)
47 \( 1 - 8.06e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.26e7iT - 6.22e13T^{2} \)
59 \( 1 - 1.46e14T^{2} \)
61 \( 1 - 1.43e7T + 1.91e14T^{2} \)
67 \( 1 + 4.06e14T^{2} \)
71 \( 1 - 6.45e14T^{2} \)
73 \( 1 + 8.06e14T^{2} \)
79 \( 1 - 6.96e7T + 1.51e15T^{2} \)
83 \( 1 + 3.84e6iT - 2.25e15T^{2} \)
89 \( 1 - 3.93e15T^{2} \)
97 \( 1 + 7.83e15T^{2} \)
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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

−13.54420653410565103323398183751, −12.03463020386680539693085394798, −11.28625091630429291855988155624, −9.425374975517169057412823621140, −8.020341811852842834617252662091, −7.35988039820324632836249891454, −6.24385332469594188633991352750, −5.22683797328678044307727658321, −2.88730279620532147633317227312, −1.26365004012369579093150663173, 0.66897873447030978731638254418, 2.49637904898507429020512654708, 3.60242279634992969322859654286, 4.88738845624000181590951079640, 6.54040448354097979665897131091, 8.374432549960790948234631982260, 9.712297620810837928519282221089, 10.36121985185948969219155767194, 11.42103550288830668724248085958, 12.20709965402453296629993409195

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