Properties

Label 2-75-5.4-c5-0-11
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $12.0287$
Root an. cond. $3.46825$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7i·2-s + 9i·3-s − 17·4-s + 63·6-s − 12i·7-s − 105i·8-s − 81·9-s + 112·11-s − 153i·12-s − 974i·13-s − 84·14-s − 1.27e3·16-s − 2.18e3i·17-s + 567i·18-s − 1.42e3·19-s + ⋯
L(s)  = 1  − 1.23i·2-s + 0.577i·3-s − 0.531·4-s + 0.714·6-s − 0.0925i·7-s − 0.580i·8-s − 0.333·9-s + 0.279·11-s − 0.306i·12-s − 1.59i·13-s − 0.114·14-s − 1.24·16-s − 1.83i·17-s + 0.412i·18-s − 0.902·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(12.0287\)
Root analytic conductor: \(3.46825\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :5/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.331765 - 1.40538i\)
\(L(\frac12)\) \(\approx\) \(0.331765 - 1.40538i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9iT \)
5 \( 1 \)
good2 \( 1 + 7iT - 32T^{2} \)
7 \( 1 + 12iT - 1.68e4T^{2} \)
11 \( 1 - 112T + 1.61e5T^{2} \)
13 \( 1 + 974iT - 3.71e5T^{2} \)
17 \( 1 + 2.18e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.42e3T + 2.47e6T^{2} \)
23 \( 1 - 3.21e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.15e3T + 2.05e7T^{2} \)
31 \( 1 + 5.68e3T + 2.86e7T^{2} \)
37 \( 1 + 6.48e3iT - 6.93e7T^{2} \)
41 \( 1 - 5.40e3T + 1.15e8T^{2} \)
43 \( 1 + 2.17e4iT - 1.47e8T^{2} \)
47 \( 1 - 368iT - 2.29e8T^{2} \)
53 \( 1 - 1.25e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.55e4T + 7.14e8T^{2} \)
61 \( 1 - 1.17e4T + 8.44e8T^{2} \)
67 \( 1 - 1.31e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.59e4T + 1.80e9T^{2} \)
73 \( 1 - 7.31e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.24e4T + 3.07e9T^{2} \)
83 \( 1 - 6.90e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.38e4T + 5.58e9T^{2} \)
97 \( 1 + 1.43e5iT - 8.58e9T^{2} \)
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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

−12.84123689825411315995857025209, −11.80640679039483869042524658167, −10.84768055715336950686340022792, −10.01542962659354872412841034912, −8.976563486114821189939478229603, −7.27004530536505098570690947726, −5.39035791730241333526925251055, −3.81251444716435884140696089486, −2.60570089969046927991988659443, −0.61791938558354349942937902987, 1.96669212965550678468147476781, 4.41296480902145491353341579345, 6.17544457407817106859297731543, 6.71778807415881387422356393942, 8.116028329715209814070841091675, 8.935428746105814864313803073967, 10.76824680821327257780463705348, 11.98742406625817617132353554742, 13.14803381755329995151211735646, 14.43350411535872721683517619118

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