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Properties

Label 2-75-15.14-c8-0-27
Degree $2$
Conductor $75$
Sign $0.842 - 0.538i$
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  + 29.5·2-s + (−69.5 − 41.5i)3-s + 614.·4-s + (−2.05e3 − 1.22e3i)6-s + 3.17e3i·7-s + 1.05e4·8-s + (3.10e3 + 5.78e3i)9-s + 1.29e4i·11-s + (−4.27e4 − 2.55e4i)12-s − 8.75e3i·13-s + 9.36e4i·14-s + 1.54e5·16-s + 1.08e5·17-s + (9.15e4 + 1.70e5i)18-s + 7.86e4·19-s + ⋯
L(s)  = 1  + 1.84·2-s + (−0.858 − 0.513i)3-s + 2.39·4-s + (−1.58 − 0.946i)6-s + 1.32i·7-s + 2.58·8-s + (0.472 + 0.881i)9-s + 0.887i·11-s + (−2.05 − 1.23i)12-s − 0.306i·13-s + 2.43i·14-s + 2.35·16-s + 1.30·17-s + (0.872 + 1.62i)18-s + 0.603·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.842 - 0.538i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.77120 + 1.39282i\)
\(L(\frac12)\) \(\approx\) \(4.77120 + 1.39282i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (69.5 + 41.5i)T \)
5 \( 1 \)
good2 \( 1 - 29.5T + 256T^{2} \)
7 \( 1 - 3.17e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.29e4iT - 2.14e8T^{2} \)
13 \( 1 + 8.75e3iT - 8.15e8T^{2} \)
17 \( 1 - 1.08e5T + 6.97e9T^{2} \)
19 \( 1 - 7.86e4T + 1.69e10T^{2} \)
23 \( 1 + 4.35e4T + 7.83e10T^{2} \)
29 \( 1 - 1.83e5iT - 5.00e11T^{2} \)
31 \( 1 - 7.80e5T + 8.52e11T^{2} \)
37 \( 1 - 2.20e6iT - 3.51e12T^{2} \)
41 \( 1 - 3.04e6iT - 7.98e12T^{2} \)
43 \( 1 + 4.84e6iT - 1.16e13T^{2} \)
47 \( 1 + 3.51e6T + 2.38e13T^{2} \)
53 \( 1 + 8.64e6T + 6.22e13T^{2} \)
59 \( 1 - 5.16e6iT - 1.46e14T^{2} \)
61 \( 1 - 5.78e6T + 1.91e14T^{2} \)
67 \( 1 + 3.67e7iT - 4.06e14T^{2} \)
71 \( 1 - 9.71e5iT - 6.45e14T^{2} \)
73 \( 1 + 9.46e6iT - 8.06e14T^{2} \)
79 \( 1 + 5.63e7T + 1.51e15T^{2} \)
83 \( 1 - 5.88e7T + 2.25e15T^{2} \)
89 \( 1 - 1.92e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.42e8iT - 7.83e15T^{2} \)
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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

−12.73303220897656206598285129155, −12.15663768869669238971865296122, −11.54000873945288419179550399061, −10.08279978976780431041759426002, −7.77838166904972219795829103469, −6.51600160235863968727966253711, −5.56158367459002540162656294707, −4.80545207146996233173964950844, −2.98789904677653460914581517974, −1.67965377449416266932111369320, 1.00477365962693063618831336582, 3.32913437593643794575862818776, 4.19231719175680208600145972503, 5.34805899863214305738262221389, 6.34699985056114957314705349020, 7.49398960577527700609612996008, 10.02205443391756798247223818093, 11.01160769936595797214735026973, 11.77495497646005795933664397710, 12.85509285551158673834402979547

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