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Properties

Label 2-471-157.156-c1-0-14
Degree $2$
Conductor $471$
Sign $0.986 - 0.163i$
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  − 0.580i·2-s + 3-s + 1.66·4-s + 2.44i·5-s − 0.580i·6-s + 1.30i·7-s − 2.12i·8-s + 9-s + 1.42·10-s − 2.31·11-s + 1.66·12-s + 2.96·13-s + 0.758·14-s + 2.44i·15-s + 2.09·16-s − 0.119·17-s + ⋯
L(s)  = 1  − 0.410i·2-s + 0.577·3-s + 0.831·4-s + 1.09i·5-s − 0.237i·6-s + 0.493i·7-s − 0.751i·8-s + 0.333·9-s + 0.449·10-s − 0.698·11-s + 0.480·12-s + 0.822·13-s + 0.202·14-s + 0.631i·15-s + 0.522·16-s − 0.0289·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.986 - 0.163i$
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: \((2,\ 471,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05162 + 0.168972i\)
\(L(\frac12)\) \(\approx\) \(2.05162 + 0.168972i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-12.3 + 2.05i)T \)
good2 \( 1 + 0.580iT - 2T^{2} \)
5 \( 1 - 2.44iT - 5T^{2} \)
7 \( 1 - 1.30iT - 7T^{2} \)
11 \( 1 + 2.31T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 + 0.119T + 17T^{2} \)
19 \( 1 + 0.543T + 19T^{2} \)
23 \( 1 - 1.90iT - 23T^{2} \)
29 \( 1 - 1.94iT - 29T^{2} \)
31 \( 1 + 2.86T + 31T^{2} \)
37 \( 1 + 2.10T + 37T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 + 5.70iT - 43T^{2} \)
47 \( 1 + 6.17T + 47T^{2} \)
53 \( 1 + 1.95iT - 53T^{2} \)
59 \( 1 + 5.11iT - 59T^{2} \)
61 \( 1 + 4.37iT - 61T^{2} \)
67 \( 1 + 1.00T + 67T^{2} \)
71 \( 1 + 9.36T + 71T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 9.78iT - 79T^{2} \)
83 \( 1 - 9.03iT - 83T^{2} \)
89 \( 1 - 0.298T + 89T^{2} \)
97 \( 1 + 5.45iT - 97T^{2} \)
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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

−10.82668378977981078540349519679, −10.49562676402745086748817452966, −9.381354824053336131499158191314, −8.289474426225223311495314386232, −7.30973767407896565415521912467, −6.58680029152577016291526895131, −5.52003515978328129791058040432, −3.69170559243093634364621589800, −2.90574993911575778213044712159, −1.92691594172969149582103607890, 1.41200970985426982591628352149, 2.86250253842428601982380413965, 4.25475113504503152259813262739, 5.36741124881044835548918866361, 6.41848238015531697170866273127, 7.49547156584635689909431763519, 8.223685704550010779170499217726, 8.930258455885944919491809202312, 10.12700881045665927786629303978, 10.93751017334090241244295289015

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