Properties

Label 2-75-15.2-c7-0-31
Degree $2$
Conductor $75$
Sign $-0.274 + 0.961i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (13.6 − 13.6i)2-s + (−31.2 + 34.8i)3-s − 244. i·4-s + (49.5 + 901. i)6-s + (896. + 896. i)7-s + (−1.59e3 − 1.59e3i)8-s + (−239. − 2.17e3i)9-s − 6.52e3i·11-s + (8.52e3 + 7.64e3i)12-s + (1.28e3 − 1.28e3i)13-s + 2.44e4·14-s − 1.22e4·16-s + (2.45e4 − 2.45e4i)17-s + (−3.29e4 − 2.64e4i)18-s + 1.64e4i·19-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (−0.667 + 0.744i)3-s − 1.91i·4-s + (0.0936 + 1.70i)6-s + (0.987 + 0.987i)7-s + (−1.10 − 1.10i)8-s + (−0.109 − 0.993i)9-s − 1.47i·11-s + (1.42 + 1.27i)12-s + (0.162 − 0.162i)13-s + 2.38·14-s − 0.746·16-s + (1.21 − 1.21i)17-s + (−1.33 − 1.06i)18-s + 0.550i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.274 + 0.961i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ -0.274 + 0.961i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.93213 - 2.55969i\)
\(L(\frac12)\) \(\approx\) \(1.93213 - 2.55969i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (31.2 - 34.8i)T \)
5 \( 1 \)
good2 \( 1 + (-13.6 + 13.6i)T - 128iT^{2} \)
7 \( 1 + (-896. - 896. i)T + 8.23e5iT^{2} \)
11 \( 1 + 6.52e3iT - 1.94e7T^{2} \)
13 \( 1 + (-1.28e3 + 1.28e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-2.45e4 + 2.45e4i)T - 4.10e8iT^{2} \)
19 \( 1 - 1.64e4iT - 8.93e8T^{2} \)
23 \( 1 + (6.37e4 + 6.37e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 7.73e4T + 1.72e10T^{2} \)
31 \( 1 - 1.45e5T + 2.75e10T^{2} \)
37 \( 1 + (4.21e4 + 4.21e4i)T + 9.49e10iT^{2} \)
41 \( 1 + 3.73e4iT - 1.94e11T^{2} \)
43 \( 1 + (-2.52e5 + 2.52e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-3.87e5 + 3.87e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-6.78e5 - 6.78e5i)T + 1.17e12iT^{2} \)
59 \( 1 - 1.92e5T + 2.48e12T^{2} \)
61 \( 1 + 2.78e5T + 3.14e12T^{2} \)
67 \( 1 + (-1.57e6 - 1.57e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 2.97e6iT - 9.09e12T^{2} \)
73 \( 1 + (2.72e6 - 2.72e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 4.57e6iT - 1.92e13T^{2} \)
83 \( 1 + (-4.26e6 - 4.26e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 5.68e6T + 4.42e13T^{2} \)
97 \( 1 + (8.27e6 + 8.27e6i)T + 8.07e13iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.30535771957858570375536836329, −11.75201018830589476039189114774, −10.98430373786697326996028885042, −9.956788520351887235411388149073, −8.473421476712520142911180521192, −5.82342200430658976207209955291, −5.32021701692550278162620744374, −3.97312181759463158095303057635, −2.69659544045218313370657461902, −0.888720390103616138394820952926, 1.55815504815060279779157933054, 4.07966274957367685171446152915, 5.06443811986691180971043904975, 6.24431458750512203706768004466, 7.46198403116496952762876737955, 7.85461336004117389447331367354, 10.31042492166116351043060599230, 11.73541401892234308999919313908, 12.61221827090076451077244902564, 13.57679752301439616028118645382

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