Properties

Label 2-75-5.2-c2-0-4
Degree $2$
Conductor $75$
Sign $0.229 + 0.973i$
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  + (−0.224 − 0.224i)2-s + (1.22 − 1.22i)3-s − 3.89i·4-s − 0.550·6-s + (−3.44 − 3.44i)7-s + (−1.77 + 1.77i)8-s − 2.99i·9-s + 11.3·11-s + (−4.77 − 4.77i)12-s + (5.55 − 5.55i)13-s + 1.55i·14-s − 14.7·16-s + (17.3 + 17.3i)17-s + (−0.674 + 0.674i)18-s + 8.69i·19-s + ⋯
L(s)  = 1  + (−0.112 − 0.112i)2-s + (0.408 − 0.408i)3-s − 0.974i·4-s − 0.0917·6-s + (−0.492 − 0.492i)7-s + (−0.221 + 0.221i)8-s − 0.333i·9-s + 1.03·11-s + (−0.397 − 0.397i)12-s + (0.426 − 0.426i)13-s + 0.110i·14-s − 0.924·16-s + (1.02 + 1.02i)17-s + (−0.0374 + 0.0374i)18-s + 0.457i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01664 - 0.804587i\)
\(L(\frac12)\) \(\approx\) \(1.01664 - 0.804587i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good2 \( 1 + (0.224 + 0.224i)T + 4iT^{2} \)
7 \( 1 + (3.44 + 3.44i)T + 49iT^{2} \)
11 \( 1 - 11.3T + 121T^{2} \)
13 \( 1 + (-5.55 + 5.55i)T - 169iT^{2} \)
17 \( 1 + (-17.3 - 17.3i)T + 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (11.5 - 11.5i)T - 529iT^{2} \)
29 \( 1 - 35.1iT - 841T^{2} \)
31 \( 1 - 10.6T + 961T^{2} \)
37 \( 1 + (-6.04 - 6.04i)T + 1.36e3iT^{2} \)
41 \( 1 - 0.696T + 1.68e3T^{2} \)
43 \( 1 + (-26.4 + 26.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.2 + 44.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (-0.696 + 0.696i)T - 2.80e3iT^{2} \)
59 \( 1 + 39.9iT - 3.48e3T^{2} \)
61 \( 1 - 5.90T + 3.72e3T^{2} \)
67 \( 1 + (-45.1 - 45.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (77.7 - 77.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (13.1 - 13.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (-24.5 - 24.5i)T + 9.40e3iT^{2} \)
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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

−14.19811006730146218905439886444, −13.10882904245762567822380954960, −11.89391511003976600665932176784, −10.52795810329223048353131285238, −9.660903649745329383884856412377, −8.402810876226042059373920286862, −6.85357534504635271669409153445, −5.75241657961393025445355604351, −3.67929523041263810703640580517, −1.36898646714398648337005601066, 2.92671815863290521531326806479, 4.27828341536192349585621990593, 6.31740806299076602911839864052, 7.70333963418304722356852670782, 8.927527622430037280535679498836, 9.669996670677761813852262975587, 11.45720818184313341912901635592, 12.26927746200339529336249430404, 13.49939100172558768235572021506, 14.49512449802776932207150238354

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