↑

Properties

Label 2-275-5.3-c2-0-2
Degree $2$
Conductor $275$
Sign $-0.437 + 0.899i$
Analytic cond. $7.49320$
Root an. cond. $2.73737$
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  + (−2.61 + 2.61i)2-s + (3.50 + 3.50i)3-s − 9.71i·4-s − 18.3·6-s + (−6.89 + 6.89i)7-s + (14.9 + 14.9i)8-s + 15.5i·9-s + 3.31·11-s + (34.0 − 34.0i)12-s + (−3.74 − 3.74i)13-s − 36.1i·14-s − 39.5·16-s + (−8.33 + 8.33i)17-s + (−40.7 − 40.7i)18-s + 24.9i·19-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)2-s + (1.16 + 1.16i)3-s − 2.42i·4-s − 3.05·6-s + (−0.985 + 0.985i)7-s + (1.87 + 1.87i)8-s + 1.72i·9-s + 0.301·11-s + (2.83 − 2.83i)12-s + (−0.287 − 0.287i)13-s − 2.58i·14-s − 2.47·16-s + (−0.490 + 0.490i)17-s + (−2.26 − 2.26i)18-s + 1.31i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(7.49320\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :1),\ -0.437 + 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.382193 - 0.611031i\)
\(L(\frac12)\) \(\approx\) \(0.382193 - 0.611031i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 - 3.31T \)
good2 \( 1 + (2.61 - 2.61i)T - 4iT^{2} \)
3 \( 1 + (-3.50 - 3.50i)T + 9iT^{2} \)
7 \( 1 + (6.89 - 6.89i)T - 49iT^{2} \)
13 \( 1 + (3.74 + 3.74i)T + 169iT^{2} \)
17 \( 1 + (8.33 - 8.33i)T - 289iT^{2} \)
19 \( 1 - 24.9iT - 361T^{2} \)
23 \( 1 + (26.1 + 26.1i)T + 529iT^{2} \)
29 \( 1 + 31.2iT - 841T^{2} \)
31 \( 1 + 2.56T + 961T^{2} \)
37 \( 1 + (7.48 - 7.48i)T - 1.36e3iT^{2} \)
41 \( 1 + 27.3T + 1.68e3T^{2} \)
43 \( 1 + (-33.0 - 33.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-17.6 + 17.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-47.0 - 47.0i)T + 2.80e3iT^{2} \)
59 \( 1 - 44.3iT - 3.48e3T^{2} \)
61 \( 1 - 0.515T + 3.72e3T^{2} \)
67 \( 1 + (-5.39 + 5.39i)T - 4.48e3iT^{2} \)
71 \( 1 - 75.9T + 5.04e3T^{2} \)
73 \( 1 + (-21.1 - 21.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 80.1iT - 6.24e3T^{2} \)
83 \( 1 + (79.3 + 79.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 + (103. - 103. i)T - 9.40e3iT^{2} \)
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

−12.28120540255939148696164348606, −10.53165836801973511018094150564, −9.916899789757662256884803162604, −9.314393828332085554989838404825, −8.517834068943079405243267665518, −7.953133005358737343644464458331, −6.47393796744915961671753023751, −5.62730427512055558272904024251, −4.07293101857826604512640134404, −2.39893812852872471666187449343, 0.46626734782640973831915094099, 1.82208544769718859376153350612, 2.93943133501159582678142849700, 3.84582563797675697607743269977, 6.94778615118404308402939259245, 7.24486041519932515692723192308, 8.391129984810436788160479680563, 9.224140818685330809083908541464, 9.799086954920469006281957143099, 10.95054580388710753442550718393

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