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Properties

Label 2-471-157.156-c1-0-17
Degree $2$
Conductor $471$
Sign $-0.143 + 0.989i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.927i·2-s − 3-s + 1.13·4-s − 0.150i·5-s + 0.927i·6-s − 1.39i·7-s − 2.91i·8-s + 9-s − 0.139·10-s + 2.55·11-s − 1.13·12-s − 1.29·13-s − 1.29·14-s + 0.150i·15-s − 0.421·16-s − 4.05·17-s + ⋯
L(s)  = 1  − 0.655i·2-s − 0.577·3-s + 0.569·4-s − 0.0673i·5-s + 0.378i·6-s − 0.526i·7-s − 1.02i·8-s + 0.333·9-s − 0.0441·10-s + 0.770·11-s − 0.329·12-s − 0.358·13-s − 0.345·14-s + 0.0388i·15-s − 0.105·16-s − 0.984·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.143 + 0.989i$
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: \((2,\ 471,\ (\ :1/2),\ -0.143 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905472 - 1.04596i\)
\(L(\frac12)\) \(\approx\) \(0.905472 - 1.04596i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 + 12.4i)T \)
good2 \( 1 + 0.927iT - 2T^{2} \)
5 \( 1 + 0.150iT - 5T^{2} \)
7 \( 1 + 1.39iT - 7T^{2} \)
11 \( 1 - 2.55T + 11T^{2} \)
13 \( 1 + 1.29T + 13T^{2} \)
17 \( 1 + 4.05T + 17T^{2} \)
19 \( 1 - 4.18T + 19T^{2} \)
23 \( 1 + 6.45iT - 23T^{2} \)
29 \( 1 + 8.65iT - 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
41 \( 1 - 3.51iT - 41T^{2} \)
43 \( 1 - 7.32iT - 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 5.69iT - 53T^{2} \)
59 \( 1 - 0.913iT - 59T^{2} \)
61 \( 1 + 5.54iT - 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 - 6.09T + 71T^{2} \)
73 \( 1 + 5.76iT - 73T^{2} \)
79 \( 1 - 15.7iT - 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 - 4.29T + 89T^{2} \)
97 \( 1 - 3.65iT - 97T^{2} \)
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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

−10.98191595792032646517556510537, −10.08185006538941621879746572540, −9.338013594833312109484584014803, −7.964070908347640015959773657332, −6.82070855823772060524329285314, −6.37593633107325456179531046432, −4.84889700821885822051665466694, −3.83974809661763254883804731196, −2.46023198491700475350146434027, −0.965405896938315116619835917184, 1.78243973300498572748421743075, 3.32192478924946522058459399672, 4.99002863788553193888971634523, 5.67174580803531906510928641262, 6.81367423984643149403570337871, 7.19288460908224088186831895083, 8.522534313778264342575697885894, 9.343267500992235151602634049871, 10.53747767869214287796966529351, 11.40073882818844593855372770275

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