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Properties

Label 2-471-471.239-c0-0-0
Degree $2$
Conductor $471$
Sign $0.325 - 0.945i$
Analytic cond. $0.235059$
Root an. cond. $0.484829$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.885 + 0.464i)3-s + (−0.885 + 0.464i)4-s + (−0.616 + 1.17i)7-s + (0.568 + 0.822i)9-s − 12-s − 0.709·13-s + (0.568 − 0.822i)16-s + (1.32 − 1.17i)19-s + (−1.09 + 0.753i)21-s + (0.970 + 0.239i)25-s + (0.120 + 0.992i)27-s − 1.32i·28-s + (−0.688 − 1.81i)31-s + (−0.885 − 0.464i)36-s + (0.180 + 1.48i)37-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)3-s + (−0.885 + 0.464i)4-s + (−0.616 + 1.17i)7-s + (0.568 + 0.822i)9-s − 12-s − 0.709·13-s + (0.568 − 0.822i)16-s + (1.32 − 1.17i)19-s + (−1.09 + 0.753i)21-s + (0.970 + 0.239i)25-s + (0.120 + 0.992i)27-s − 1.32i·28-s + (−0.688 − 1.81i)31-s + (−0.885 − 0.464i)36-s + (0.180 + 1.48i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.325 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8983174567\)
\(L(\frac12)\) \(\approx\) \(0.8983174567\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.885 - 0.464i)T \)
157 \( 1 + (-0.885 - 0.464i)T \)
good2 \( 1 + (0.885 - 0.464i)T^{2} \)
5 \( 1 + (-0.970 - 0.239i)T^{2} \)
7 \( 1 + (0.616 - 1.17i)T + (-0.568 - 0.822i)T^{2} \)
11 \( 1 + (-0.885 - 0.464i)T^{2} \)
13 \( 1 + 0.709T + T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 + (-1.32 + 1.17i)T + (0.120 - 0.992i)T^{2} \)
23 \( 1 + (-0.354 - 0.935i)T^{2} \)
29 \( 1 + (-0.970 + 0.239i)T^{2} \)
31 \( 1 + (0.688 + 1.81i)T + (-0.748 + 0.663i)T^{2} \)
37 \( 1 + (-0.180 - 1.48i)T + (-0.970 + 0.239i)T^{2} \)
41 \( 1 + (-0.354 + 0.935i)T^{2} \)
43 \( 1 + (0.764 + 1.45i)T + (-0.568 + 0.822i)T^{2} \)
47 \( 1 + (0.970 + 0.239i)T^{2} \)
53 \( 1 + (0.120 - 0.992i)T^{2} \)
59 \( 1 + (0.120 + 0.992i)T^{2} \)
61 \( 1 + (-0.317 - 0.358i)T + (-0.120 + 0.992i)T^{2} \)
67 \( 1 + (0.850 - 0.753i)T + (0.120 - 0.992i)T^{2} \)
71 \( 1 + (-0.120 + 0.992i)T^{2} \)
73 \( 1 + (-0.922 + 1.75i)T + (-0.568 - 0.822i)T^{2} \)
79 \( 1 + (-0.222 + 0.902i)T + (-0.885 - 0.464i)T^{2} \)
83 \( 1 + (-0.748 + 0.663i)T^{2} \)
89 \( 1 + (0.748 - 0.663i)T^{2} \)
97 \( 1 + (1.97 + 0.239i)T + (0.970 + 0.239i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.55181509433573355187943226367, −10.12115527889374716528882155171, −9.329908185636215792323434177545, −9.012474008153097586574203676021, −8.009219318425817066261872984333, −7.07685317373483924160806862161, −5.45828378212751013797492219106, −4.67754636980944872119214197033, −3.37143441410539665159214472636, −2.58585408423146059612916902917, 1.26210362603075135177926397394, 3.19243124714162449362933693156, 4.06317577187486867897551825372, 5.28412792198638399944547864074, 6.66035602996337025869791734528, 7.45683228720846276970652901319, 8.362033094285564922020425788887, 9.431758072680671891442979907228, 9.900107559604573398592804840863, 10.76068051183152275955271825872

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