↑

Properties

Label 2-154-11.5-c1-0-5
Degree $2$
Conductor $154$
Sign $-0.437 + 0.899i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (−0.309 − 0.951i)2-s + (−0.5 − 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.618 − 1.90i)5-s + (−0.190 + 0.587i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 2.48i)9-s − 1.99·10-s + (−3.04 − 1.31i)11-s + 0.618·12-s + (−0.381 − 1.17i)13-s + (−0.809 − 0.587i)14-s + (−1 + 0.726i)15-s + (0.309 − 0.951i)16-s + (0.263 − 0.812i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.288 − 0.209i)3-s + (−0.404 + 0.293i)4-s + (0.276 − 0.850i)5-s + (−0.0779 + 0.239i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.829i)9-s − 0.632·10-s + (−0.918 − 0.396i)11-s + 0.178·12-s + (−0.105 − 0.326i)13-s + (−0.216 − 0.157i)14-s + (−0.258 + 0.187i)15-s + (0.0772 − 0.237i)16-s + (0.0640 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.466298 - 0.745770i\)
\(L(\frac12)\) \(\approx\) \(0.466298 - 0.745770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (3.04 + 1.31i)T \)
good3 \( 1 + (0.5 + 0.363i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.618 + 1.90i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.381 + 1.17i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.263 + 0.812i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.54 - 4.02i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.472T + 23T^{2} \)
29 \( 1 + (-6.47 + 4.70i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.38 - 4.25i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.61 - 3.35i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.73 - 3.44i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (-9.47 - 6.88i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4 + 12.3i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.78 - 6.37i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.763 - 2.35i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + (-0.472 + 1.45i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.73 + 2.71i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1 - 3.07i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.95 + 15.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 0.326T + 89T^{2} \)
97 \( 1 + (3.04 + 9.37i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.41652942329558184710390647938, −11.83313498165413430350901469402, −10.63220161376494828459532161445, −9.666847599135427447376037989977, −8.626032926727745947114646897389, −7.62677702051659098767023098909, −5.90260068799733650700026680225, −4.82990760273024257856527022806, −3.13571364566150972936328767804, −1.04725796057058880884171440520, 2.60374388067666243876877972287, 4.75582238432853706319910556332, 5.69030176823974327248486304361, 7.02055330562513519206322120720, 7.87811550807892099801914168491, 9.169739219638960014396150217635, 10.37879560633075023694019842448, 10.94619296098511763329752998824, 12.26973755948007580242539111638, 13.70207990341289899046647921606

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