Properties

Label 2-75-15.2-c5-0-17
Degree $2$
Conductor $75$
Sign $0.229 + 0.973i$
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  + (−1.22 + 1.22i)2-s + (−11.0 − 11.0i)3-s + 29i·4-s + 26.9·6-s + (−74.7 − 74.7i)8-s + 242. i·9-s + (319. − 319. i)12-s − 745·16-s + (1.62e3 − 1.62e3i)17-s + (−297. − 297. i)18-s − 2.16e3i·19-s + (−3.45e3 − 3.45e3i)23-s + 1.64e3i·24-s + (2.67e3 − 2.67e3i)27-s + 8.15e3·31-s + (3.30e3 − 3.30e3i)32-s + ⋯
L(s)  = 1  + (−0.216 + 0.216i)2-s + (−0.707 − 0.707i)3-s + 0.906i·4-s + 0.306·6-s + (−0.412 − 0.412i)8-s + i·9-s + (0.640 − 0.640i)12-s − 0.727·16-s + (1.36 − 1.36i)17-s + (−0.216 − 0.216i)18-s − 1.37i·19-s + (−1.36 − 1.36i)23-s + 0.583i·24-s + (0.707 − 0.707i)27-s + 1.52·31-s + (0.570 − 0.570i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(12.0287\)
Root analytic conductor: \(3.46825\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :5/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.660992 - 0.523121i\)
\(L(\frac12)\) \(\approx\) \(0.660992 - 0.523121i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (11.0 + 11.0i)T \)
5 \( 1 \)
good2 \( 1 + (1.22 - 1.22i)T - 32iT^{2} \)
7 \( 1 + 1.68e4iT^{2} \)
11 \( 1 - 1.61e5T^{2} \)
13 \( 1 - 3.71e5iT^{2} \)
17 \( 1 + (-1.62e3 + 1.62e3i)T - 1.41e6iT^{2} \)
19 \( 1 + 2.16e3iT - 2.47e6T^{2} \)
23 \( 1 + (3.45e3 + 3.45e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 2.05e7T^{2} \)
31 \( 1 - 8.15e3T + 2.86e7T^{2} \)
37 \( 1 + 6.93e7iT^{2} \)
41 \( 1 - 1.15e8T^{2} \)
43 \( 1 - 1.47e8iT^{2} \)
47 \( 1 + (1.96e4 - 1.96e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-284. - 284. i)T + 4.18e8iT^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 - 3.48e4T + 8.44e8T^{2} \)
67 \( 1 + 1.35e9iT^{2} \)
71 \( 1 - 1.80e9T^{2} \)
73 \( 1 - 2.07e9iT^{2} \)
79 \( 1 + 7.00e4iT - 3.07e9T^{2} \)
83 \( 1 + (7.25e4 + 7.25e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 + 8.58e9iT^{2} \)
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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

−13.15408331803662400901693555144, −12.14133984561096516538948132799, −11.49242156193870570203306503835, −9.940733329820229580163891125779, −8.400395557732094386339150552223, −7.41771739610932976842406398687, −6.39163235979324175720705534518, −4.75094301488601034777153184951, −2.72448235849727979760634240531, −0.46942097395760816159121276717, 1.36530432482200037396190724604, 3.81084782390231776237438994288, 5.44657938468985319099827460879, 6.20488896473297388638427146390, 8.210747697144371775479682168056, 9.901862648173814374069129823218, 10.13379511779155538835789539103, 11.44979955965501680692242399102, 12.33665250067552161227255828942, 14.05754576502471397223914473080

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