↑

Properties

Label 2-750-5.3-c2-0-19
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + 2.44·6-s + (6.47 − 6.47i)7-s + (2 + 2i)8-s + 2.99i·9-s + 0.239·11-s + (−2.44 + 2.44i)12-s + (3.35 + 3.35i)13-s + 12.9i·14-s − 4·16-s + (14.3 − 14.3i)17-s + (−2.99 − 2.99i)18-s − 2.54i·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 + (0.924 − 0.924i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + 0.0217·11-s + (−0.204 + 0.204i)12-s + (0.258 + 0.258i)13-s + 0.924i·14-s − 0.250·16-s + (0.844 − 0.844i)17-s + (−0.166 − 0.166i)18-s − 0.134i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.346063616\)
\(L(\frac12)\) \(\approx\) \(1.346063616\)
\(L(2)\) 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 \( 1 + (-6.47 + 6.47i)T - 49iT^{2} \)
11 \( 1 - 0.239T + 121T^{2} \)
13 \( 1 + (-3.35 - 3.35i)T + 169iT^{2} \)
17 \( 1 + (-14.3 + 14.3i)T - 289iT^{2} \)
19 \( 1 + 2.54iT - 361T^{2} \)
23 \( 1 + (-26.7 - 26.7i)T + 529iT^{2} \)
29 \( 1 - 19.8iT - 841T^{2} \)
31 \( 1 + 33.8T + 961T^{2} \)
37 \( 1 + (-25.5 + 25.5i)T - 1.36e3iT^{2} \)
41 \( 1 + 31.3T + 1.68e3T^{2} \)
43 \( 1 + (52.7 + 52.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-19.7 + 19.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-32.6 - 32.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 42.2iT - 3.48e3T^{2} \)
61 \( 1 - 47.6T + 3.72e3T^{2} \)
67 \( 1 + (-86.4 + 86.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 128.T + 5.04e3T^{2} \)
73 \( 1 + (-21.2 - 21.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 53.3iT - 6.24e3T^{2} \)
83 \( 1 + (81.9 + 81.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 1.92iT - 7.92e3T^{2} \)
97 \( 1 + (37.1 - 37.1i)T - 9.40e3iT^{2} \)
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

−10.02502395806813741425404245185, −9.120322958407578381222880901680, −8.153128151213085515473536259428, −7.28635800103325352643640526519, −6.94498177988440232608486939631, −5.52746825739132302367724854167, −4.95087436579595490772884458255, −3.58003301937736253191562682736, −1.74958368989616928897065745287, −0.71808830605227645368318861244, 1.14540388481235238744268868942, 2.44672515161386295722895547705, 3.68717361584778685053625128576, 4.86437997061941210211314783440, 5.66656009042613508040214422043, 6.76237229879279437570621655705, 8.116727757055862399677460954015, 8.491033380563616183227071152782, 9.520408534068133484102978906280, 10.30216938509432611608429438093

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