↑

Properties

Label 2-153-153.11-c1-0-0
Degree $2$
Conductor $153$
Sign $-0.532 - 0.846i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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  + (−0.491 − 0.641i)2-s + (−1.73 − 0.0616i)3-s + (0.348 − 1.30i)4-s + (−3.10 − 0.203i)5-s + (0.812 + 1.14i)6-s + (0.330 + 5.03i)7-s + (−2.49 + 1.03i)8-s + (2.99 + 0.213i)9-s + (1.39 + 2.09i)10-s + (0.620 + 0.543i)11-s + (−0.683 + 2.23i)12-s + (−4.35 − 1.16i)13-s + (3.06 − 2.68i)14-s + (5.36 + 0.544i)15-s + (−0.439 − 0.253i)16-s + (−3.78 + 1.62i)17-s + ⋯
L(s)  = 1  + (−0.347 − 0.453i)2-s + (−0.999 − 0.0355i)3-s + (0.174 − 0.650i)4-s + (−1.38 − 0.0910i)5-s + (0.331 + 0.465i)6-s + (0.124 + 1.90i)7-s + (−0.883 + 0.365i)8-s + (0.997 + 0.0711i)9-s + (0.442 + 0.661i)10-s + (0.186 + 0.163i)11-s + (−0.197 + 0.643i)12-s + (−1.20 − 0.323i)13-s + (0.819 − 0.718i)14-s + (1.38 + 0.140i)15-s + (−0.109 − 0.0634i)16-s + (−0.918 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.532 - 0.846i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ -0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0541702 + 0.0980327i\)
\(L(\frac12)\) \(\approx\) \(0.0541702 + 0.0980327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0616i)T \)
17 \( 1 + (3.78 - 1.62i)T \)
good2 \( 1 + (0.491 + 0.641i)T + (-0.517 + 1.93i)T^{2} \)
5 \( 1 + (3.10 + 0.203i)T + (4.95 + 0.652i)T^{2} \)
7 \( 1 + (-0.330 - 5.03i)T + (-6.94 + 0.913i)T^{2} \)
11 \( 1 + (-0.620 - 0.543i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (4.35 + 1.16i)T + (11.2 + 6.5i)T^{2} \)
19 \( 1 + (1.95 + 0.811i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.27 + 3.75i)T + (-18.2 + 14.0i)T^{2} \)
29 \( 1 + (0.376 - 0.185i)T + (17.6 - 23.0i)T^{2} \)
31 \( 1 + (-0.875 - 0.997i)T + (-4.04 + 30.7i)T^{2} \)
37 \( 1 + (-0.0132 - 0.0664i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.49 - 3.03i)T + (-24.9 - 32.5i)T^{2} \)
43 \( 1 + (-0.945 + 7.17i)T + (-41.5 - 11.1i)T^{2} \)
47 \( 1 + (-5.19 + 1.39i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.70 - 6.52i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.70 - 2.07i)T + (15.2 + 56.9i)T^{2} \)
61 \( 1 + (1.71 - 0.112i)T + (60.4 - 7.96i)T^{2} \)
67 \( 1 + (-1.90 + 1.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.14 + 0.426i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-4.79 - 3.20i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (8.73 - 9.96i)T + (-10.3 - 78.3i)T^{2} \)
83 \( 1 + (-2.91 + 2.23i)T + (21.4 - 80.1i)T^{2} \)
89 \( 1 + (-3.26 + 3.26i)T - 89iT^{2} \)
97 \( 1 + (7.70 + 15.6i)T + (-59.0 + 76.9i)T^{2} \)
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

−12.55579666855536863108987263465, −12.10517407385019901856278059508, −11.46375151161172470180743164009, −10.55492145852882428459090496528, −9.322125179535131581951425056634, −8.313830547522934722300177127016, −6.80024524917110073677011067369, −5.64769813068704992929984324311, −4.60510031441731250312130063482, −2.34900520282817117173279599828, 0.13025874741519588135431229501, 3.79654564902860757534203212875, 4.53504156313230080304912482018, 6.70178983575682585054858635929, 7.29221030998669266807354599325, 7.963238341496359655204618606188, 9.652077465655337179577980936569, 10.90490618955604656301266177575, 11.54533851305612557559681762898, 12.39075473239613536621729768044

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