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Properties

Label 2-75-5.4-c7-0-0
Degree $2$
Conductor $75$
Sign $0.447 + 0.894i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.1i·2-s + 27i·3-s − 101.·4-s − 408.·6-s + 198. i·7-s + 407. i·8-s − 729·9-s − 5.26e3·11-s − 2.72e3i·12-s − 1.21e3i·13-s − 3.01e3·14-s − 1.91e4·16-s − 3.45e4i·17-s − 1.10e4i·18-s − 1.86e4·19-s + ⋯
L(s)  = 1  + 1.33i·2-s + 0.577i·3-s − 0.789·4-s − 0.772·6-s + 0.219i·7-s + 0.281i·8-s − 0.333·9-s − 1.19·11-s − 0.455i·12-s − 0.153i·13-s − 0.293·14-s − 1.16·16-s − 1.70i·17-s − 0.445i·18-s − 0.622·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.225636 - 0.139451i\)
\(L(\frac12)\) \(\approx\) \(0.225636 - 0.139451i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27iT \)
5 \( 1 \)
good2 \( 1 - 15.1iT - 128T^{2} \)
7 \( 1 - 198. iT - 8.23e5T^{2} \)
11 \( 1 + 5.26e3T + 1.94e7T^{2} \)
13 \( 1 + 1.21e3iT - 6.27e7T^{2} \)
17 \( 1 + 3.45e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.86e4T + 8.93e8T^{2} \)
23 \( 1 - 3.33e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.78e5T + 1.72e10T^{2} \)
31 \( 1 + 2.37e5T + 2.75e10T^{2} \)
37 \( 1 - 4.82e5iT - 9.49e10T^{2} \)
41 \( 1 - 2.93e5T + 1.94e11T^{2} \)
43 \( 1 + 4.43e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.81e4iT - 5.06e11T^{2} \)
53 \( 1 + 1.66e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.75e6T + 2.48e12T^{2} \)
61 \( 1 + 3.15e6T + 3.14e12T^{2} \)
67 \( 1 - 2.29e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.71e6T + 9.09e12T^{2} \)
73 \( 1 - 2.67e6iT - 1.10e13T^{2} \)
79 \( 1 + 3.44e6T + 1.92e13T^{2} \)
83 \( 1 + 1.71e6iT - 2.71e13T^{2} \)
89 \( 1 + 3.52e6T + 4.42e13T^{2} \)
97 \( 1 - 1.44e7iT - 8.07e13T^{2} \)
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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

−14.27725979099622140692698234731, −13.32762292629111400690742315675, −11.75017472126840584845126196379, −10.52465115487057917496196747526, −9.204269018106329150062148569955, −8.103363065944355400649924285767, −7.03510198594618798712833283773, −5.62074543651933436627331023046, −4.77447491652459104401299138416, −2.72147096715287104997604370064, 0.085392835771727571337748015367, 1.59894080764473113331254024960, 2.75597472497793872879218896709, 4.24925815523485444179559158555, 6.11024958504832026711128529619, 7.61724819395374531984633734425, 8.926492247153388406286236946505, 10.49203853106878573050615137612, 10.87401463145050654114399131888, 12.45797416320093234086572849633

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