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Properties

Label 2-750-5.3-c2-0-16
Degree $2$
Conductor $750$
Sign $0.707 - 0.707i$
Analytic cond. $20.4360$
Root an. cond. $4.52062$
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  + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s − 2.44·6-s + (0.244 − 0.244i)7-s + (2 + 2i)8-s + 2.99i·9-s + 16.9·11-s + (2.44 − 2.44i)12-s + (2.02 + 2.02i)13-s + 0.489i·14-s − 4·16-s + (−1.80 + 1.80i)17-s + (−2.99 − 2.99i)18-s − 31.8i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s − 0.408·6-s + (0.0349 − 0.0349i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + 1.53·11-s + (0.204 − 0.204i)12-s + (0.155 + 0.155i)13-s + 0.0349i·14-s − 0.250·16-s + (−0.106 + 0.106i)17-s + (−0.166 − 0.166i)18-s − 1.67i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(20.4360\)
Root analytic conductor: \(4.52062\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.827014943\)
\(L(\frac12)\) \(\approx\) \(1.827014943\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-0.244 + 0.244i)T - 49iT^{2} \)
11 \( 1 - 16.9T + 121T^{2} \)
13 \( 1 + (-2.02 - 2.02i)T + 169iT^{2} \)
17 \( 1 + (1.80 - 1.80i)T - 289iT^{2} \)
19 \( 1 + 31.8iT - 361T^{2} \)
23 \( 1 + (-2.22 - 2.22i)T + 529iT^{2} \)
29 \( 1 + 21.6iT - 841T^{2} \)
31 \( 1 - 15.2T + 961T^{2} \)
37 \( 1 + (1.00 - 1.00i)T - 1.36e3iT^{2} \)
41 \( 1 + 1.49T + 1.68e3T^{2} \)
43 \( 1 + (-54.0 - 54.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-37.1 + 37.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-58.5 - 58.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 85.9iT - 3.48e3T^{2} \)
61 \( 1 + 11.9T + 3.72e3T^{2} \)
67 \( 1 + (33.8 - 33.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 81.7T + 5.04e3T^{2} \)
73 \( 1 + (0.0883 + 0.0883i)T + 5.32e3iT^{2} \)
79 \( 1 - 99.3iT - 6.24e3T^{2} \)
83 \( 1 + (11.2 + 11.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 38.7iT - 7.92e3T^{2} \)
97 \( 1 + (57.8 - 57.8i)T - 9.40e3iT^{2} \)
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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

−9.988055751604490894156885260937, −9.167723586291804308765715572797, −8.827122460237706565074428855487, −7.70775308043207598182608922829, −6.82405217630314752859884407009, −6.05131513354687521781301473049, −4.75476857648666249787859633481, −3.93472341863536271137018091461, −2.51099442967117748765788058733, −0.995279096798155159494122482874, 1.01850585801379704675770565611, 2.03494167294549534982636784437, 3.40919258640734054749987322726, 4.16422902974960195346270345862, 5.75667538588934760813818481132, 6.73084614700412013078609913821, 7.57266928939532981336944385237, 8.548726965769443390819415268033, 9.087429643394421175829922620936, 9.998259343060371563693058674383

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