↑

Properties

Label 2-151-151.64-c1-0-4
Degree $2$
Conductor $151$
Sign $0.207 + 0.978i$
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  − 0.877·2-s + (−2.41 + 1.75i)3-s − 1.23·4-s + (0.174 − 0.538i)5-s + (2.11 − 1.53i)6-s + (0.391 − 1.20i)7-s + 2.83·8-s + (1.82 − 5.62i)9-s + (−0.153 + 0.471i)10-s + (−2.00 − 1.45i)11-s + (2.97 − 2.15i)12-s + (2.03 − 1.48i)13-s + (−0.343 + 1.05i)14-s + (0.521 + 1.60i)15-s − 0.0239·16-s + (−1.41 − 4.34i)17-s + ⋯
L(s)  = 1  − 0.620·2-s + (−1.39 + 1.01i)3-s − 0.615·4-s + (0.0781 − 0.240i)5-s + (0.864 − 0.628i)6-s + (0.148 − 0.455i)7-s + 1.00·8-s + (0.608 − 1.87i)9-s + (−0.0484 + 0.149i)10-s + (−0.604 − 0.439i)11-s + (0.857 − 0.623i)12-s + (0.565 − 0.410i)13-s + (−0.0918 + 0.282i)14-s + (0.134 + 0.414i)15-s − 0.00598·16-s + (−0.342 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $0.207 + 0.978i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ 0.207 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218433 - 0.176965i\)
\(L(\frac12)\) \(\approx\) \(0.218433 - 0.176965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (12.2 - 0.819i)T \)
good2 \( 1 + 0.877T + 2T^{2} \)
3 \( 1 + (2.41 - 1.75i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.174 + 0.538i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.391 + 1.20i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (2.00 + 1.45i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.03 + 1.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.41 + 4.34i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + 4.63T + 19T^{2} \)
23 \( 1 + 0.156T + 23T^{2} \)
29 \( 1 + (6.90 + 5.01i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.71 + 5.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.30 + 4.58i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.39 - 2.46i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.656 - 2.02i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-7.82 + 5.68i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.23 + 5.98i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + (3.62 + 2.63i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.66 - 11.2i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.75 + 8.47i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.0542 - 0.166i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.263 - 0.812i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.0633 + 0.194i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-0.0418 - 0.128i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.32 + 16.3i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.76338689537974140143912471894, −11.21463418799070091972764299412, −10.82664474754532306195854367372, −9.816372819127290795072059409074, −8.985847764731538891078696117781, −7.63183799989929873765681639334, −5.98896963943461899859268323009, −4.97708433429859294351281913416, −4.03276850558013157968089167662, −0.41966785633250458554173846327, 1.68234804773565054020465539131, 4.57824945490797820637042049226, 5.77493655703131660029836973483, 6.81440183274851762185146516794, 7.946904255896135885002322192143, 8.997491290001193453792778400573, 10.59785460915517973513129184206, 10.92109198421718131737532108007, 12.43339690140943764177391783978, 12.83635966908534427595878642270

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