Properties

Label 2-75-25.16-c3-0-8
Degree $2$
Conductor $75$
Sign $0.0300 - 0.999i$
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  + (3.87 + 2.81i)2-s + (0.927 + 2.85i)3-s + (4.62 + 14.2i)4-s + (7.51 − 8.27i)5-s + (−4.44 + 13.6i)6-s + 0.140·7-s + (−10.3 + 31.7i)8-s + (−7.28 + 5.29i)9-s + (52.4 − 10.9i)10-s + (−38.3 − 27.8i)11-s + (−36.3 + 26.4i)12-s + (−27.7 + 20.1i)13-s + (0.544 + 0.395i)14-s + (30.5 + 13.7i)15-s + (−32.6 + 23.7i)16-s + (5.10 − 15.7i)17-s + ⋯
L(s)  = 1  + (1.37 + 0.996i)2-s + (0.178 + 0.549i)3-s + (0.578 + 1.78i)4-s + (0.672 − 0.740i)5-s + (−0.302 + 0.930i)6-s + 0.00758·7-s + (−0.456 + 1.40i)8-s + (−0.269 + 0.195i)9-s + (1.65 − 0.344i)10-s + (−1.05 − 0.762i)11-s + (−0.874 + 0.635i)12-s + (−0.592 + 0.430i)13-s + (0.0104 + 0.00755i)14-s + (0.526 + 0.237i)15-s + (−0.510 + 0.371i)16-s + (0.0728 − 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.0300 - 0.999i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.0300 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.26712 + 2.19999i\)
\(L(\frac12)\) \(\approx\) \(2.26712 + 2.19999i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.927 - 2.85i)T \)
5 \( 1 + (-7.51 + 8.27i)T \)
good2 \( 1 + (-3.87 - 2.81i)T + (2.47 + 7.60i)T^{2} \)
7 \( 1 - 0.140T + 343T^{2} \)
11 \( 1 + (38.3 + 27.8i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (27.7 - 20.1i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-5.10 + 15.7i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-28.1 + 86.5i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (-130. - 94.9i)T + (3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (3.81 + 11.7i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (102. - 316. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-227. + 165. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (101. - 73.8i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 529.T + 7.95e4T^{2} \)
47 \( 1 + (-23.1 - 71.3i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (77.0 + 237. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-217. + 157. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (299. + 217. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (54.5 - 167. i)T + (-2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (-223. - 686. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-1.97 - 1.43i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-187. - 577. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (392. - 1.20e3i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + (-1.08e3 - 786. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (442. + 1.36e3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−14.23411964300107052466853074059, −13.44089982351058717773931269155, −12.72061671378552870172237769061, −11.26859825299121428153648101330, −9.635442662624531555676814670422, −8.362327335740259639769029939716, −6.93344788815647230254985836662, −5.38843510346771740963041177454, −4.87630716321674703617331713130, −3.07762679665932953268665232022, 2.02818071962507131032258080366, 3.10815793662132313578097325818, 4.98966178624132675253971082342, 6.18014876009845989087887528782, 7.65431943965144880279718850468, 9.856667625404306173757011646244, 10.64830475294740989223260819114, 11.85411578730770860723032281122, 12.94965874536873563083120247233, 13.38977452018162162246357266102

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