Properties

Label 2-75-15.8-c11-0-1
Degree $2$
Conductor $75$
Sign $-0.991 + 0.130i$
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  + (297. + 297. i)3-s − 2.04e3i·4-s + (−3.16e4 + 3.16e4i)7-s + 1.77e5i·9-s + (6.09e5 − 6.09e5i)12-s + (1.02e6 + 1.02e6i)13-s − 4.19e6·16-s − 2.05e7i·19-s − 1.88e7·21-s + (−5.27e7 + 5.27e7i)27-s + (6.47e7 + 6.47e7i)28-s − 2.96e8·31-s + 3.62e8·36-s + (−2.22e8 + 2.22e8i)37-s + 6.11e8i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − i·4-s + (−0.710 + 0.710i)7-s + i·9-s + (0.707 − 0.707i)12-s + (0.767 + 0.767i)13-s − 16-s − 1.90i·19-s − 1.00·21-s + (−0.707 + 0.707i)27-s + (0.710 + 0.710i)28-s − 1.85·31-s + 36-s + (−0.526 + 0.526i)37-s + 1.08i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.991 + 0.130i$
Analytic conductor: \(57.6257\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :11/2),\ -0.991 + 0.130i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0162829 - 0.247963i\)
\(L(\frac12)\) \(\approx\) \(0.0162829 - 0.247963i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-297. - 297. i)T \)
5 \( 1 \)
good2 \( 1 + 2.04e3iT^{2} \)
7 \( 1 + (3.16e4 - 3.16e4i)T - 1.97e9iT^{2} \)
11 \( 1 - 2.85e11T^{2} \)
13 \( 1 + (-1.02e6 - 1.02e6i)T + 1.79e12iT^{2} \)
17 \( 1 + 3.42e13iT^{2} \)
19 \( 1 + 2.05e7iT - 1.16e14T^{2} \)
23 \( 1 - 9.52e14iT^{2} \)
29 \( 1 + 1.22e16T^{2} \)
31 \( 1 + 2.96e8T + 2.54e16T^{2} \)
37 \( 1 + (2.22e8 - 2.22e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 5.50e17T^{2} \)
43 \( 1 + (5.43e8 + 5.43e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 2.47e18iT^{2} \)
53 \( 1 - 9.26e18iT^{2} \)
59 \( 1 + 3.01e19T^{2} \)
61 \( 1 + 1.29e10T + 4.35e19T^{2} \)
67 \( 1 + (1.50e10 - 1.50e10i)T - 1.22e20iT^{2} \)
71 \( 1 - 2.31e20T^{2} \)
73 \( 1 + (2.07e10 + 2.07e10i)T + 3.13e20iT^{2} \)
79 \( 1 - 3.28e10iT - 7.47e20T^{2} \)
83 \( 1 - 1.28e21iT^{2} \)
89 \( 1 + 2.77e21T^{2} \)
97 \( 1 + (-1.13e11 + 1.13e11i)T - 7.15e21iT^{2} \)
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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

−13.15059089927752186958875168341, −11.43173684581408935284702325119, −10.49836487857073040869930543704, −9.241071921918636970894334920492, −8.926004912860731650088509541770, −6.96278640965055431313490996628, −5.68736782786914442550082332390, −4.50315886648988168895441079517, −3.01541324804076927839993440163, −1.77216418948185215635305155796, 0.05279509049021667045244243105, 1.62489561597490699178297349571, 3.22870021691568552496723756488, 3.80367994291385343671087913458, 6.09474984546190884697324785816, 7.32486095404885809005014099712, 8.064949537659199132445725878084, 9.174625482117987520846614831732, 10.54683452553301899325690081686, 12.08267599671356877451938289992

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