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Properties

Label 2-75-5.3-c8-0-10
Degree $2$
Conductor $75$
Sign $0.991 - 0.130i$
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  + (17.7 − 17.7i)2-s + (33.0 + 33.0i)3-s − 373. i·4-s + 1.17e3·6-s + (−3.27e3 + 3.27e3i)7-s + (−2.09e3 − 2.09e3i)8-s + 2.18e3i·9-s + 1.78e4·11-s + (1.23e4 − 1.23e4i)12-s + (2.10e4 + 2.10e4i)13-s + 1.16e5i·14-s + 2.14e4·16-s + (1.10e4 − 1.10e4i)17-s + (3.88e4 + 3.88e4i)18-s − 4.19e3i·19-s + ⋯
L(s)  = 1  + (1.10 − 1.10i)2-s + (0.408 + 0.408i)3-s − 1.46i·4-s + 0.905·6-s + (−1.36 + 1.36i)7-s + (−0.510 − 0.510i)8-s + 0.333i·9-s + 1.21·11-s + (0.596 − 0.596i)12-s + (0.736 + 0.736i)13-s + 3.02i·14-s + 0.327·16-s + (0.132 − 0.132i)17-s + (0.369 + 0.369i)18-s − 0.0321i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.77823 + 0.248103i\)
\(L(\frac12)\) \(\approx\) \(3.77823 + 0.248103i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.0 - 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (-17.7 + 17.7i)T - 256iT^{2} \)
7 \( 1 + (3.27e3 - 3.27e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.78e4T + 2.14e8T^{2} \)
13 \( 1 + (-2.10e4 - 2.10e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-1.10e4 + 1.10e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 4.19e3iT - 1.69e10T^{2} \)
23 \( 1 + (-1.94e5 - 1.94e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 7.28e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.11e6T + 8.52e11T^{2} \)
37 \( 1 + (2.27e6 - 2.27e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 5.27e6T + 7.98e12T^{2} \)
43 \( 1 + (1.31e6 + 1.31e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (4.96e5 - 4.96e5i)T - 2.38e13iT^{2} \)
53 \( 1 + (1.05e6 + 1.05e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 2.25e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.64e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.76e7 + 1.76e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 7.32e6T + 6.45e14T^{2} \)
73 \( 1 + (-1.61e7 - 1.61e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 1.60e7iT - 1.51e15T^{2} \)
83 \( 1 + (-6.51e7 - 6.51e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 1.05e8iT - 3.93e15T^{2} \)
97 \( 1 + (2.43e7 - 2.43e7i)T - 7.83e15iT^{2} \)
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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

−12.76996240420755240074387877315, −11.99667859540422825804466066900, −11.01364259929156338491224940722, −9.548154170868129510305253361992, −8.935769115699279784570339948476, −6.54323616448550707370950693199, −5.35094914182062736222907981100, −3.79493842881963760085795764354, −3.04867542813073782626134928915, −1.67186562886994867486469991363, 0.817952671575187585229262844476, 3.38352576874984977685818523979, 4.08936909160903774746902181926, 6.00505124227554519650880043219, 6.78299782370551912381752953172, 7.65816344790654245823157385333, 9.232297737942846958061458411085, 10.64736747628961151945662465636, 12.49380714865526592521991887398, 13.15496499791379870518722007004

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