Properties

Label 2-153-153.113-c1-0-1
Degree $2$
Conductor $153$
Sign $0.885 - 0.463i$
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  + (−0.999 + 0.131i)2-s + (−0.711 − 1.57i)3-s + (−0.949 + 0.254i)4-s + (1.57 + 3.18i)5-s + (0.919 + 1.48i)6-s + (−0.0400 − 0.0197i)7-s + (2.77 − 1.15i)8-s + (−1.98 + 2.24i)9-s + (−1.99 − 2.98i)10-s + (5.72 − 1.94i)11-s + (1.07 + 1.31i)12-s + (0.940 + 3.51i)13-s + (0.0426 + 0.0144i)14-s + (3.91 − 4.75i)15-s + (−0.924 + 0.533i)16-s + (3.77 + 1.66i)17-s + ⋯
L(s)  = 1  + (−0.706 + 0.0930i)2-s + (−0.410 − 0.911i)3-s + (−0.474 + 0.127i)4-s + (0.703 + 1.42i)5-s + (0.375 + 0.606i)6-s + (−0.0151 − 0.00746i)7-s + (0.982 − 0.407i)8-s + (−0.662 + 0.749i)9-s + (−0.629 − 0.942i)10-s + (1.72 − 0.585i)11-s + (0.311 + 0.380i)12-s + (0.260 + 0.973i)13-s + (0.0113 + 0.00386i)14-s + (1.01 − 1.22i)15-s + (−0.231 + 0.133i)16-s + (0.914 + 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.885 - 0.463i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ 0.885 - 0.463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.696170 + 0.171281i\)
\(L(\frac12)\) \(\approx\) \(0.696170 + 0.171281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.711 + 1.57i)T \)
17 \( 1 + (-3.77 - 1.66i)T \)
good2 \( 1 + (0.999 - 0.131i)T + (1.93 - 0.517i)T^{2} \)
5 \( 1 + (-1.57 - 3.18i)T + (-3.04 + 3.96i)T^{2} \)
7 \( 1 + (0.0400 + 0.0197i)T + (4.26 + 5.55i)T^{2} \)
11 \( 1 + (-5.72 + 1.94i)T + (8.72 - 6.69i)T^{2} \)
13 \( 1 + (-0.940 - 3.51i)T + (-11.2 + 6.5i)T^{2} \)
19 \( 1 + (2.46 + 1.02i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.75 - 5.42i)T + (-3.00 - 22.8i)T^{2} \)
29 \( 1 + (-0.00257 + 0.0392i)T + (-28.7 - 3.78i)T^{2} \)
31 \( 1 + (-1.33 + 3.94i)T + (-24.5 - 18.8i)T^{2} \)
37 \( 1 + (1.01 + 5.09i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-4.78 + 0.313i)T + (40.6 - 5.35i)T^{2} \)
43 \( 1 + (0.651 + 0.500i)T + (11.1 + 41.5i)T^{2} \)
47 \( 1 + (0.435 - 1.62i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.17 - 2.83i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.31 + 9.99i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-0.0408 + 0.0827i)T + (-37.1 - 48.3i)T^{2} \)
67 \( 1 + (-5.04 - 2.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.65 - 1.12i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (6.75 + 4.51i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-1.72 - 5.09i)T + (-62.6 + 48.0i)T^{2} \)
83 \( 1 + (0.760 + 5.77i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-12.7 + 12.7i)T - 89iT^{2} \)
97 \( 1 + (3.62 + 0.237i)T + (96.1 + 12.6i)T^{2} \)
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

−13.22644381668370643767275855391, −11.84048121859059269724771947645, −11.05427410735695049144343270009, −9.930168290665394739446958406850, −8.981042962472055016616772975966, −7.70598981857135291364330935539, −6.68843962466384179724226089320, −5.97222145502648207806186080208, −3.76647549729751429911479299936, −1.70174955597260354082570527217, 1.13689655677333896268262885495, 4.13940700762695470493564704265, 5.03359413022411258323479167807, 6.14454184704340503083512319268, 8.262722271680013514484282032243, 9.024358357833610690159527413674, 9.743133793644246243097101544644, 10.41353681577038622507937655927, 11.92826562627890633428229677898, 12.69371773512448738067229128941

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