Properties

Label 2-1150-115.22-c1-0-1
Degree $2$
Conductor $1150$
Sign $-0.498 - 0.866i$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (1.22 − 1.22i)3-s + 1.00i·4-s − 1.73·6-s + (−1.50 + 1.50i)7-s + (0.707 − 0.707i)8-s − 5.03i·11-s + (1.22 + 1.22i)12-s + (−3.34 + 3.34i)13-s + 2.12·14-s − 1.00·16-s + (−5.06 + 5.06i)17-s − 5.03·19-s + 3.68i·21-s + (−3.56 + 3.56i)22-s + (−2.91 − 3.80i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.707 − 0.707i)3-s + 0.500i·4-s − 0.707·6-s + (−0.569 + 0.569i)7-s + (0.250 − 0.250i)8-s − 1.51i·11-s + (0.353 + 0.353i)12-s + (−0.928 + 0.928i)13-s + 0.569·14-s − 0.250·16-s + (−1.22 + 1.22i)17-s − 1.15·19-s + 0.804i·21-s + (−0.759 + 0.759i)22-s + (−0.607 − 0.794i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.498 - 0.866i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ -0.498 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09014913268\)
\(L(\frac12)\) \(\approx\) \(0.09014913268\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
23 \( 1 + (2.91 + 3.80i)T \)
good3 \( 1 + (-1.22 + 1.22i)T - 3iT^{2} \)
7 \( 1 + (1.50 - 1.50i)T - 7iT^{2} \)
11 \( 1 + 5.03iT - 11T^{2} \)
13 \( 1 + (3.34 - 3.34i)T - 13iT^{2} \)
17 \( 1 + (5.06 - 5.06i)T - 17iT^{2} \)
19 \( 1 + 5.03T + 19T^{2} \)
29 \( 1 + 5.46iT - 29T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 + (4.11 - 4.11i)T - 37iT^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + (-1.10 - 1.10i)T + 43iT^{2} \)
47 \( 1 + (3.34 + 3.34i)T + 47iT^{2} \)
53 \( 1 + (9.73 + 9.73i)T + 53iT^{2} \)
59 \( 1 + 0.535iT - 59T^{2} \)
61 \( 1 - 3.68iT - 61T^{2} \)
67 \( 1 + (-2.05 + 2.05i)T - 67iT^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 + (-0.568 + 0.568i)T - 73iT^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + (1.10 - 1.10i)T - 97iT^{2} \)
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

−10.04621844918524141954247896164, −9.009348480827511830611252190602, −8.433565123291351949881927424230, −8.069286636419673865796616368874, −6.58591990118403917413252528408, −6.38551559127811720785036203786, −4.74462873688725444258169524708, −3.59610393444627790717437682372, −2.50723911342600226815870484252, −1.89905913669511466711614726012, 0.03800909347313345600066192224, 2.12294245595762346096391460851, 3.24732940249303505327312108653, 4.43055131021268097978890082067, 5.01366736487567886572519837615, 6.51250331935515237201602857715, 7.08915092261120935273242631950, 7.88223583698667773919939208416, 8.906957086093588950535979929020, 9.496926851503714260599368060746

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