Properties

Label 2-75-5.4-c11-0-27
Degree $2$
Conductor $75$
Sign $-0.447 - 0.894i$
Analytic cond. $57.6257$
Root an. cond. $7.59116$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 51.6i·2-s − 243i·3-s − 617.·4-s − 1.25e4·6-s + 7.43e4i·7-s − 7.38e4i·8-s − 5.90e4·9-s + 7.15e5·11-s + 1.49e5i·12-s − 1.87e6i·13-s + 3.84e6·14-s − 5.07e6·16-s + 8.90e5i·17-s + 3.04e6i·18-s − 1.87e7·19-s + ⋯
L(s)  = 1  − 1.14i·2-s − 0.577i·3-s − 0.301·4-s − 0.658·6-s + 1.67i·7-s − 0.797i·8-s − 0.333·9-s + 1.33·11-s + 0.173i·12-s − 1.40i·13-s + 1.90·14-s − 1.21·16-s + 0.152i·17-s + 0.380i·18-s − 1.73·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(57.6257\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :11/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.414589 + 0.670820i\)
\(L(\frac12)\) \(\approx\) \(0.414589 + 0.670820i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243iT \)
5 \( 1 \)
good2 \( 1 + 51.6iT - 2.04e3T^{2} \)
7 \( 1 - 7.43e4iT - 1.97e9T^{2} \)
11 \( 1 - 7.15e5T + 2.85e11T^{2} \)
13 \( 1 + 1.87e6iT - 1.79e12T^{2} \)
17 \( 1 - 8.90e5iT - 3.42e13T^{2} \)
19 \( 1 + 1.87e7T + 1.16e14T^{2} \)
23 \( 1 - 5.63e5iT - 9.52e14T^{2} \)
29 \( 1 + 1.17e8T + 1.22e16T^{2} \)
31 \( 1 + 1.96e7T + 2.54e16T^{2} \)
37 \( 1 + 6.18e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.30e9T + 5.50e17T^{2} \)
43 \( 1 + 4.05e7iT - 9.29e17T^{2} \)
47 \( 1 + 1.59e9iT - 2.47e18T^{2} \)
53 \( 1 - 1.39e9iT - 9.26e18T^{2} \)
59 \( 1 + 6.33e8T + 3.01e19T^{2} \)
61 \( 1 - 2.84e9T + 4.35e19T^{2} \)
67 \( 1 - 4.48e9iT - 1.22e20T^{2} \)
71 \( 1 - 1.58e10T + 2.31e20T^{2} \)
73 \( 1 + 4.54e9iT - 3.13e20T^{2} \)
79 \( 1 + 1.60e10T + 7.47e20T^{2} \)
83 \( 1 - 3.01e10iT - 1.28e21T^{2} \)
89 \( 1 - 2.70e10T + 2.77e21T^{2} \)
97 \( 1 + 1.16e11iT - 7.15e21T^{2} \)
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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

−11.71572695588414185527941601424, −10.68161277832193142601011299448, −9.315179796866782674995000269238, −8.414870492358842924328624777843, −6.68920887230833274413415192143, −5.62346995182442940623245140376, −3.70861001885624266408387213625, −2.46579881438609618105317069496, −1.65821916869663844473458328863, −0.18324995935846829263684994629, 1.65324492160299753020677229151, 3.86462276264681132308614065652, 4.63828371042549302577865702655, 6.45406425928533507752687771564, 6.94320983756435185333775320026, 8.334415366256530093311121100580, 9.471751958318564382635650545446, 10.78592710419087335441042010985, 11.65971144242467210079695947858, 13.50697634094301944315245649651

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