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Properties

Label 2-75-5.4-c7-0-10
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  − 18.3i·2-s + 27i·3-s − 208.·4-s + 495.·6-s + 254. i·7-s + 1.47e3i·8-s − 729·9-s + 7.73e3·11-s − 5.62e3i·12-s + 9.88e3i·13-s + 4.66e3·14-s + 405.·16-s − 3.11e4i·17-s + 1.33e4i·18-s + 2.46e4·19-s + ⋯
L(s)  = 1  − 1.62i·2-s + 0.577i·3-s − 1.62·4-s + 0.936·6-s + 0.280i·7-s + 1.01i·8-s − 0.333·9-s + 1.75·11-s − 0.940i·12-s + 1.24i·13-s + 0.454·14-s + 0.0247·16-s − 1.53i·17-s + 0.540i·18-s + 0.824·19-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

\(L(4)\) \(\approx\) \(0.973988 - 1.57594i\)
\(L(\frac12)\) \(\approx\) \(0.973988 - 1.57594i\)
\(L(\frac{9}{2})\) 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 \( 1 + 18.3iT - 128T^{2} \)
7 \( 1 - 254. iT - 8.23e5T^{2} \)
11 \( 1 - 7.73e3T + 1.94e7T^{2} \)
13 \( 1 - 9.88e3iT - 6.27e7T^{2} \)
17 \( 1 + 3.11e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.46e4T + 8.93e8T^{2} \)
23 \( 1 + 9.91e4iT - 3.40e9T^{2} \)
29 \( 1 - 5.94e4T + 1.72e10T^{2} \)
31 \( 1 + 4.73e4T + 2.75e10T^{2} \)
37 \( 1 + 1.03e4iT - 9.49e10T^{2} \)
41 \( 1 - 2.20e5T + 1.94e11T^{2} \)
43 \( 1 + 9.60e5iT - 2.71e11T^{2} \)
47 \( 1 - 3.31e5iT - 5.06e11T^{2} \)
53 \( 1 - 6.54e5iT - 1.17e12T^{2} \)
59 \( 1 + 3.76e5T + 2.48e12T^{2} \)
61 \( 1 - 1.28e6T + 3.14e12T^{2} \)
67 \( 1 + 1.96e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.21e6T + 9.09e12T^{2} \)
73 \( 1 + 3.46e6iT - 1.10e13T^{2} \)
79 \( 1 - 4.06e6T + 1.92e13T^{2} \)
83 \( 1 + 5.25e6iT - 2.71e13T^{2} \)
89 \( 1 - 3.76e6T + 4.42e13T^{2} \)
97 \( 1 - 1.05e7iT - 8.07e13T^{2} \)
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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

−12.07534043486101991283648870524, −11.80377267026864928139558024189, −10.67418525635100514197103381648, −9.342141418514584858969059180136, −9.063881918462651335532805044387, −6.72187762807536842119074753353, −4.72862721821124896299045984492, −3.72919844202300299688097064132, −2.32339640311532990791017918346, −0.802615048747830999464336084643, 1.16112182015189229124002557271, 3.78329588678387560748788091899, 5.52675373796703179707600640617, 6.44749609553849265777635972962, 7.50278203704172279814392400712, 8.418428203032976871736883160616, 9.634420832832216142441419879637, 11.38369992463526813813838541478, 12.74295531707029524107561240319, 13.78462615035675644579613069725

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