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Properties

Label 2-75-5.4-c9-0-4
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  + 38.7i·2-s + 81i·3-s − 992.·4-s − 3.14e3·6-s + 1.22e4i·7-s − 1.86e4i·8-s − 6.56e3·9-s − 6.58e4·11-s − 8.03e4i·12-s + 1.12e5i·13-s − 4.75e5·14-s + 2.14e5·16-s + 9.66e4i·17-s − 2.54e5i·18-s + 1.81e5·19-s + ⋯
L(s)  = 1  + 1.71i·2-s + 0.577i·3-s − 1.93·4-s − 0.989·6-s + 1.93i·7-s − 1.60i·8-s − 0.333·9-s − 1.35·11-s − 1.11i·12-s + 1.09i·13-s − 3.31·14-s + 0.818·16-s + 0.280i·17-s − 0.571i·18-s + 0.319·19-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

\(L(5)\) \(\approx\) \(0.955331 - 0.225523i\)
\(L(\frac12)\) \(\approx\) \(0.955331 - 0.225523i\)
\(L(\frac{11}{2})\) 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 \( 1 - 38.7iT - 512T^{2} \)
7 \( 1 - 1.22e4iT - 4.03e7T^{2} \)
11 \( 1 + 6.58e4T + 2.35e9T^{2} \)
13 \( 1 - 1.12e5iT - 1.06e10T^{2} \)
17 \( 1 - 9.66e4iT - 1.18e11T^{2} \)
19 \( 1 - 1.81e5T + 3.22e11T^{2} \)
23 \( 1 + 2.44e5iT - 1.80e12T^{2} \)
29 \( 1 - 5.26e6T + 1.45e13T^{2} \)
31 \( 1 - 1.83e6T + 2.64e13T^{2} \)
37 \( 1 - 6.36e6iT - 1.29e14T^{2} \)
41 \( 1 - 1.57e6T + 3.27e14T^{2} \)
43 \( 1 - 1.99e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.00e7iT - 1.11e15T^{2} \)
53 \( 1 - 2.57e6iT - 3.29e15T^{2} \)
59 \( 1 - 1.19e8T + 8.66e15T^{2} \)
61 \( 1 - 1.92e8T + 1.16e16T^{2} \)
67 \( 1 + 1.20e8iT - 2.72e16T^{2} \)
71 \( 1 + 6.99e5T + 4.58e16T^{2} \)
73 \( 1 - 8.91e7iT - 5.88e16T^{2} \)
79 \( 1 + 4.31e8T + 1.19e17T^{2} \)
83 \( 1 - 1.69e7iT - 1.86e17T^{2} \)
89 \( 1 - 3.09e6T + 3.50e17T^{2} \)
97 \( 1 - 5.72e8iT - 7.60e17T^{2} \)
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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

−14.17038541062893807552579771601, −12.86508287786997755309195619126, −11.58780660546604508460705704399, −9.821303332299468002302099005436, −8.788897106427554398606071864197, −8.100474032553013780017361395578, −6.49281417684530619765763004953, −5.52081291131769634854249212791, −4.69644514636486380669805167969, −2.59892370285273864769782292847, 0.34921553033541188286077896034, 0.988666208855612042954269633854, 2.57145220611146377447161179544, 3.69157841466815926839114600967, 5.06134848634657493680892311398, 7.19736457276668087501201360164, 8.269319783872599733219641871785, 10.14664876916128426306783834497, 10.43258995243896439417410621377, 11.54007922852535907555880197250

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