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Properties

Label 2-75-5.4-c9-0-0
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + 81i·3-s + 496·4-s − 324·6-s + 7.68e3i·7-s + 4.03e3i·8-s − 6.56e3·9-s − 8.64e4·11-s + 4.01e4i·12-s − 1.49e5i·13-s − 3.07e4·14-s + 2.37e5·16-s + 2.07e5i·17-s − 2.62e4i·18-s − 7.16e5·19-s + ⋯
L(s)  = 1  + 0.176i·2-s + 0.577i·3-s + 0.968·4-s − 0.102·6-s + 1.20i·7-s + 0.348i·8-s − 0.333·9-s − 1.77·11-s + 0.559i·12-s − 1.45i·13-s − 0.213·14-s + 0.907·16-s + 0.602i·17-s − 0.0589i·18-s − 1.26·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.158314 - 0.670631i\)
\(L(\frac12)\) \(\approx\) \(0.158314 - 0.670631i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81iT \)
5 \( 1 \)
good2 \( 1 - 4iT - 512T^{2} \)
7 \( 1 - 7.68e3iT - 4.03e7T^{2} \)
11 \( 1 + 8.64e4T + 2.35e9T^{2} \)
13 \( 1 + 1.49e5iT - 1.06e10T^{2} \)
17 \( 1 - 2.07e5iT - 1.18e11T^{2} \)
19 \( 1 + 7.16e5T + 3.22e11T^{2} \)
23 \( 1 - 1.36e6iT - 1.80e12T^{2} \)
29 \( 1 - 3.19e6T + 1.45e13T^{2} \)
31 \( 1 + 2.34e6T + 2.64e13T^{2} \)
37 \( 1 + 1.87e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.92e7T + 3.27e14T^{2} \)
43 \( 1 + 1.51e6iT - 5.02e14T^{2} \)
47 \( 1 + 6.15e5iT - 1.11e15T^{2} \)
53 \( 1 - 4.74e6iT - 3.29e15T^{2} \)
59 \( 1 + 6.06e7T + 8.66e15T^{2} \)
61 \( 1 + 1.26e8T + 1.16e16T^{2} \)
67 \( 1 - 1.11e8iT - 2.72e16T^{2} \)
71 \( 1 + 1.75e8T + 4.58e16T^{2} \)
73 \( 1 + 6.12e7iT - 5.88e16T^{2} \)
79 \( 1 + 2.34e8T + 1.19e17T^{2} \)
83 \( 1 - 1.18e8iT - 1.86e17T^{2} \)
89 \( 1 - 3.16e8T + 3.50e17T^{2} \)
97 \( 1 + 2.42e8iT - 7.60e17T^{2} \)
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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

−13.06488928321643429260425222349, −12.21389048669177698784253749386, −10.84736229267042311470074822995, −10.27663886282752811415848033816, −8.584138866640782052052020482067, −7.69974410833557688044115780508, −5.93897706999291239208798759484, −5.26769795600549132133849368098, −3.12368918391922915203187438378, −2.18559328528314018581411905691, 0.17019188866220638551600038178, 1.69885216145436342025879757835, 2.86480430232243717305632710423, 4.61960758604826487511586179250, 6.44966173437567963351896827818, 7.20093600452806536895084319698, 8.291560109421426462985219891314, 10.19578272533152834210059261165, 10.88071072894415545891633281119, 12.00782136959570479769930608390

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