Properties

Label 2-151-151.58-c1-0-0
Degree $2$
Conductor $151$
Sign $-0.729 - 0.684i$
Analytic cond. $1.20574$
Root an. cond. $1.09806$
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  + (−0.963 + 0.429i)2-s + (0.840 + 1.78i)3-s + (−0.593 + 0.659i)4-s + (1.24 + 3.35i)5-s + (−1.57 − 1.36i)6-s + (−3.37 − 2.23i)7-s + (0.941 − 2.89i)8-s + (−0.570 + 0.690i)9-s + (−2.64 − 2.69i)10-s + (−1.19 − 1.71i)11-s + (−1.67 − 0.506i)12-s + (1.96 + 2.59i)13-s + (4.20 + 0.711i)14-s + (−4.94 + 5.04i)15-s + (0.150 + 1.43i)16-s + (4.30 + 2.13i)17-s + ⋯
L(s)  = 1  + (−0.681 + 0.303i)2-s + (0.485 + 1.03i)3-s + (−0.296 + 0.329i)4-s + (0.557 + 1.50i)5-s + (−0.643 − 0.555i)6-s + (−1.27 − 0.846i)7-s + (0.332 − 1.02i)8-s + (−0.190 + 0.230i)9-s + (−0.835 − 0.853i)10-s + (−0.359 − 0.517i)11-s + (−0.483 − 0.146i)12-s + (0.545 + 0.719i)13-s + (1.12 + 0.190i)14-s + (−1.27 + 1.30i)15-s + (0.0375 + 0.357i)16-s + (1.04 + 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $-0.729 - 0.684i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ -0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297512 + 0.751689i\)
\(L(\frac12)\) \(\approx\) \(0.297512 + 0.751689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (4.48 - 11.4i)T \)
good2 \( 1 + (0.963 - 0.429i)T + (1.33 - 1.48i)T^{2} \)
3 \( 1 + (-0.840 - 1.78i)T + (-1.91 + 2.31i)T^{2} \)
5 \( 1 + (-1.24 - 3.35i)T + (-3.78 + 3.26i)T^{2} \)
7 \( 1 + (3.37 + 2.23i)T + (2.71 + 6.45i)T^{2} \)
11 \( 1 + (1.19 + 1.71i)T + (-3.83 + 10.3i)T^{2} \)
13 \( 1 + (-1.96 - 2.59i)T + (-3.49 + 12.5i)T^{2} \)
17 \( 1 + (-4.30 - 2.13i)T + (10.2 + 13.5i)T^{2} \)
19 \( 1 + (-1.42 - 4.37i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.75 + 0.798i)T + (21.0 + 9.35i)T^{2} \)
29 \( 1 + (-2.34 + 1.28i)T + (15.5 - 24.4i)T^{2} \)
31 \( 1 + (-0.658 - 1.26i)T + (-17.6 + 25.4i)T^{2} \)
37 \( 1 + (2.57 - 6.12i)T + (-25.8 - 26.4i)T^{2} \)
41 \( 1 + (-1.99 + 10.4i)T + (-38.1 - 15.0i)T^{2} \)
43 \( 1 + (-4.76 + 3.16i)T + (16.6 - 39.6i)T^{2} \)
47 \( 1 + (3.74 - 1.30i)T + (36.8 - 29.1i)T^{2} \)
53 \( 1 + (3.68 + 3.46i)T + (3.32 + 52.8i)T^{2} \)
59 \( 1 + (-8.05 + 5.85i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.74 + 0.230i)T + (60.1 - 10.1i)T^{2} \)
67 \( 1 + (8.04 + 9.72i)T + (-12.5 + 65.8i)T^{2} \)
71 \( 1 + (-12.2 + 6.09i)T + (42.9 - 56.5i)T^{2} \)
73 \( 1 + (0.283 - 4.50i)T + (-72.4 - 9.14i)T^{2} \)
79 \( 1 + (-4.58 - 7.22i)T + (-33.6 + 71.4i)T^{2} \)
83 \( 1 + (7.52 - 2.97i)T + (60.5 - 56.8i)T^{2} \)
89 \( 1 + (-0.630 - 0.499i)T + (20.3 + 86.6i)T^{2} \)
97 \( 1 + (6.51 - 1.10i)T + (91.6 - 31.9i)T^{2} \)
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

−13.87492028672480920547641462706, −12.49434673129667492990905165767, −10.69509312767185545001456184411, −10.02484534966536521817161793025, −9.672245726209285787801007003114, −8.320912899781065056493449718173, −7.07127074645836198359826975898, −6.18468782693902031107797689247, −3.78926879515768314779207643681, −3.33941647654401953368482055655, 1.03365360451469215147700915207, 2.53983817664531946245837316030, 5.06006190907576096153268991786, 6.02017549294711352271118809400, 7.72058551126592443504921739436, 8.665497704932149549566304072907, 9.407454675998245257010879301921, 10.12033569761131459513257436504, 11.92993850371070052084537984523, 12.93826002029833439776976410581

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