Properties

Label 2-75-15.14-c8-0-11
Degree $2$
Conductor $75$
Sign $0.561 - 0.827i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 24.1·2-s + (70.6 − 39.5i)3-s + 326.·4-s + (−1.70e3 + 955. i)6-s − 1.04e3i·7-s − 1.70e3·8-s + (3.42e3 − 5.59e3i)9-s + 1.95e4i·11-s + (2.30e4 − 1.29e4i)12-s + 2.90e4i·13-s + 2.51e4i·14-s − 4.24e4·16-s + 1.22e5·17-s + (−8.27e4 + 1.35e5i)18-s − 1.89e5·19-s + ⋯
L(s)  = 1  − 1.50·2-s + (0.872 − 0.488i)3-s + 1.27·4-s + (−1.31 + 0.737i)6-s − 0.434i·7-s − 0.415·8-s + (0.522 − 0.852i)9-s + 1.33i·11-s + (1.11 − 0.623i)12-s + 1.01i·13-s + 0.654i·14-s − 0.648·16-s + 1.46·17-s + (−0.788 + 1.28i)18-s − 1.45·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.869434 + 0.460458i\)
\(L(\frac12)\) \(\approx\) \(0.869434 + 0.460458i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-70.6 + 39.5i)T \)
5 \( 1 \)
good2 \( 1 + 24.1T + 256T^{2} \)
7 \( 1 + 1.04e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.95e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.90e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.22e5T + 6.97e9T^{2} \)
19 \( 1 + 1.89e5T + 1.69e10T^{2} \)
23 \( 1 + 1.12e5T + 7.83e10T^{2} \)
29 \( 1 + 1.08e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.19e6T + 8.52e11T^{2} \)
37 \( 1 - 2.84e6iT - 3.51e12T^{2} \)
41 \( 1 - 3.90e6iT - 7.98e12T^{2} \)
43 \( 1 + 8.64e5iT - 1.16e13T^{2} \)
47 \( 1 - 1.48e6T + 2.38e13T^{2} \)
53 \( 1 - 3.65e6T + 6.22e13T^{2} \)
59 \( 1 - 1.46e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.60e7T + 1.91e14T^{2} \)
67 \( 1 + 2.14e7iT - 4.06e14T^{2} \)
71 \( 1 - 4.10e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.70e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.43e7T + 1.51e15T^{2} \)
83 \( 1 + 6.88e6T + 2.25e15T^{2} \)
89 \( 1 + 3.39e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.50e7iT - 7.83e15T^{2} \)
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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

−12.97982902553364259811044832637, −11.82551681702061705381949299813, −10.26358968319007273352499604898, −9.582504245613595251609170297866, −8.503416929123685231275513988345, −7.52558185802016554809352978102, −6.72624890275671927963294012520, −4.17147970617198534720506466201, −2.20571303905321534613618725268, −1.23324189755459361500511501713, 0.49232574244417781463516169509, 2.13386673392249412346519329104, 3.55787929991437081308220669203, 5.65055269364736295026034273375, 7.53781109987521173105364258756, 8.404738964120927162236640229486, 9.089643106708177449320264883163, 10.29066089776247821925772283563, 10.93509909187073698627453014324, 12.65183766747729652205097337018

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