↑

Properties

Label 2-471-157.156-c1-0-0
Degree $2$
Conductor $471$
Sign $-0.479 + 0.877i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
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  + 2.06i·2-s − 3-s − 2.27·4-s − 1.58i·5-s − 2.06i·6-s + 2.74i·7-s − 0.565i·8-s + 9-s + 3.27·10-s − 2.88·11-s + 2.27·12-s − 5.68·13-s − 5.68·14-s + 1.58i·15-s − 3.37·16-s + 1.55·17-s + ⋯
L(s)  = 1  + 1.46i·2-s − 0.577·3-s − 1.13·4-s − 0.708i·5-s − 0.843i·6-s + 1.03i·7-s − 0.200i·8-s + 0.333·9-s + 1.03·10-s − 0.869·11-s + 0.656·12-s − 1.57·13-s − 1.51·14-s + 0.408i·15-s − 0.844·16-s + 0.377·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.479 + 0.877i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ -0.479 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205387 - 0.346420i\)
\(L(\frac12)\) \(\approx\) \(0.205387 - 0.346420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
157 \( 1 + (-6.01 + 10.9i)T \)
good2 \( 1 - 2.06iT - 2T^{2} \)
5 \( 1 + 1.58iT - 5T^{2} \)
7 \( 1 - 2.74iT - 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
13 \( 1 + 5.68T + 13T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 + 2.86T + 19T^{2} \)
23 \( 1 - 0.736iT - 23T^{2} \)
29 \( 1 + 3.04iT - 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + 5.70T + 37T^{2} \)
41 \( 1 - 0.515iT - 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 2.51iT - 53T^{2} \)
59 \( 1 - 3.36iT - 59T^{2} \)
61 \( 1 - 4.10iT - 61T^{2} \)
67 \( 1 + 3.78T + 67T^{2} \)
71 \( 1 + 9.38T + 71T^{2} \)
73 \( 1 - 1.99iT - 73T^{2} \)
79 \( 1 - 9.92iT - 79T^{2} \)
83 \( 1 + 4.11iT - 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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

−11.84972121807245730319003806081, −10.64288872031579657260968070497, −9.487200456023488975086695179950, −8.744515437638724343508823177844, −7.80510098817947814370987356476, −7.06684702649139823629701144928, −5.85165678715543466018804298265, −5.31019630062355125556832174630, −4.59326701149698118815231296943, −2.40041962998123373739270027229, 0.24625223820744735602653944343, 2.06001157314582030095818967929, 3.20484360188028947256720406755, 4.30690009856745827285519001240, 5.35008187721713670546410436235, 6.97706150429599959636126430105, 7.40198023415679232709518966727, 9.044361097301594409066896404572, 10.22074494101426287550842215690, 10.47358611244345744365281113652

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