Properties

Label 2-75-25.17-c2-0-5
Degree $2$
Conductor $75$
Sign $0.816 + 0.577i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.421 − 0.827i)2-s + (0.270 − 1.71i)3-s + (1.84 + 2.53i)4-s + (4.83 + 1.28i)5-s + (−1.30 − 0.945i)6-s + (−4.18 − 4.18i)7-s + (6.54 − 1.03i)8-s + (−2.85 − 0.927i)9-s + (3.10 − 3.45i)10-s + (0.527 + 1.62i)11-s + (4.84 − 2.46i)12-s + (−7.76 − 15.2i)13-s + (−5.22 + 1.69i)14-s + (3.51 − 7.91i)15-s + (−1.97 + 6.08i)16-s + (3.52 + 22.2i)17-s + ⋯
L(s)  = 1  + (0.210 − 0.413i)2-s + (0.0903 − 0.570i)3-s + (0.461 + 0.634i)4-s + (0.966 + 0.257i)5-s + (−0.216 − 0.157i)6-s + (−0.597 − 0.597i)7-s + (0.818 − 0.129i)8-s + (−0.317 − 0.103i)9-s + (0.310 − 0.345i)10-s + (0.0479 + 0.147i)11-s + (0.403 − 0.205i)12-s + (−0.597 − 1.17i)13-s + (−0.373 + 0.121i)14-s + (0.234 − 0.527i)15-s + (−0.123 + 0.380i)16-s + (0.207 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57349 - 0.500592i\)
\(L(\frac12)\) \(\approx\) \(1.57349 - 0.500592i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.270 + 1.71i)T \)
5 \( 1 + (-4.83 - 1.28i)T \)
good2 \( 1 + (-0.421 + 0.827i)T + (-2.35 - 3.23i)T^{2} \)
7 \( 1 + (4.18 + 4.18i)T + 49iT^{2} \)
11 \( 1 + (-0.527 - 1.62i)T + (-97.8 + 71.1i)T^{2} \)
13 \( 1 + (7.76 + 15.2i)T + (-99.3 + 136. i)T^{2} \)
17 \( 1 + (-3.52 - 22.2i)T + (-274. + 89.3i)T^{2} \)
19 \( 1 + (12.3 - 16.9i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + (20.2 + 10.3i)T + (310. + 427. i)T^{2} \)
29 \( 1 + (-13.5 - 18.6i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (44.0 + 32.0i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (21.0 - 10.7i)T + (804. - 1.10e3i)T^{2} \)
41 \( 1 + (-2.79 + 8.60i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 + (-14.3 + 14.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-49.3 - 7.82i)T + (2.10e3 + 682. i)T^{2} \)
53 \( 1 + (-10.7 + 67.5i)T + (-2.67e3 - 868. i)T^{2} \)
59 \( 1 + (51.3 + 16.6i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-23.2 - 71.6i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + (0.398 + 2.51i)T + (-4.26e3 + 1.38e3i)T^{2} \)
71 \( 1 + (-98.0 + 71.2i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-38.1 - 19.4i)T + (3.13e3 + 4.31e3i)T^{2} \)
79 \( 1 + (-36.7 - 50.5i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (57.8 - 9.15i)T + (6.55e3 - 2.12e3i)T^{2} \)
89 \( 1 + (-4.08 + 1.32i)T + (6.40e3 - 4.65e3i)T^{2} \)
97 \( 1 + (170. + 27.0i)T + (8.94e3 + 2.90e3i)T^{2} \)
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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

−13.91296928139390166887475333843, −12.78294823694170498423723859023, −12.50964857153990916361683839395, −10.73979666856248435950371305158, −10.06937823178646387832909248400, −8.243792044439812871243054761421, −7.08566561594873063283349998329, −5.91533240729649929338038814851, −3.62159433978064177016772238845, −2.08768923922362323666509284188, 2.35438524526158557021534040776, 4.81184971851680932021157277176, 5.89619181933351684548332240352, 7.00787991415143255686385734974, 9.114327414621387220670904692295, 9.704066348925424331082860450174, 10.94778469307058172377202526779, 12.23369618581935830953011835584, 13.77517668758795559164201655067, 14.31096357864637288187092672438

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