Properties

Label 2-75-15.8-c3-0-8
Degree $2$
Conductor $75$
Sign $0.973 + 0.229i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.67 + 3.67i)3-s − 8i·4-s + (22.0 − 22.0i)7-s + 27i·9-s + (29.3 − 29.3i)12-s + (44.0 + 44.0i)13-s − 64·16-s − 56i·19-s + 162·21-s + (−99.2 + 99.2i)27-s + (−176. − 176. i)28-s − 308·31-s + 216·36-s + (−308. + 308. i)37-s + 324i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − i·4-s + (1.19 − 1.19i)7-s + i·9-s + (0.707 − 0.707i)12-s + (0.940 + 0.940i)13-s − 16-s − 0.676i·19-s + 1.68·21-s + (−0.707 + 0.707i)27-s + (−1.19 − 1.19i)28-s − 1.78·31-s + 36-s + (−1.37 + 1.37i)37-s + 1.33i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96702 - 0.229028i\)
\(L(\frac12)\) \(\approx\) \(1.96702 - 0.229028i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.67 - 3.67i)T \)
5 \( 1 \)
good2 \( 1 + 8iT^{2} \)
7 \( 1 + (-22.0 + 22.0i)T - 343iT^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + (-44.0 - 44.0i)T + 2.19e3iT^{2} \)
17 \( 1 + 4.91e3iT^{2} \)
19 \( 1 + 56iT - 6.85e3T^{2} \)
23 \( 1 - 1.21e4iT^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + 308T + 2.97e4T^{2} \)
37 \( 1 + (308. - 308. i)T - 5.06e4iT^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + (-154. - 154. i)T + 7.95e4iT^{2} \)
47 \( 1 + 1.03e5iT^{2} \)
53 \( 1 - 1.48e5iT^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 182T + 2.26e5T^{2} \)
67 \( 1 + (-462. + 462. i)T - 3.00e5iT^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (-264. - 264. i)T + 3.89e5iT^{2} \)
79 \( 1 - 884iT - 4.93e5T^{2} \)
83 \( 1 - 5.71e5iT^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + (969. - 969. i)T - 9.12e5iT^{2} \)
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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

−14.08507481607727206377388415741, −13.53218601281134818583959912024, −11.20698112847302956502300289720, −10.75990262272355757801903590000, −9.536167660706677099755196055713, −8.428831700997192317029759745771, −6.99512621823117408623741034430, −5.10951789879178630656387643032, −4.03940896076538959345397896946, −1.61366602085828627509237671951, 2.04844984145428904494340941490, 3.57917861674967906275874497601, 5.65164556792110258427606561309, 7.41399193393128351082120012605, 8.342099669103551740273421157770, 8.958559805647251843468297307401, 11.07257257710514082809060607739, 12.18171412002587779672094195421, 12.83270081223020830181334983656, 14.04474504100760711320545544311

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