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Properties

Label 2-471-157.156-c3-0-2
Degree $2$
Conductor $471$
Sign $-0.849 - 0.527i$
Analytic cond. $27.7898$
Root an. cond. $5.27161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.783i·2-s − 3·3-s + 7.38·4-s − 11.1i·5-s − 2.35i·6-s − 5.77i·7-s + 12.0i·8-s + 9·9-s + 8.73·10-s − 23.4·11-s − 22.1·12-s − 81.0·13-s + 4.52·14-s + 33.4i·15-s + 49.6·16-s − 42.8·17-s + ⋯
L(s)  = 1  + 0.277i·2-s − 0.577·3-s + 0.923·4-s − 0.996i·5-s − 0.159i·6-s − 0.311i·7-s + 0.532i·8-s + 0.333·9-s + 0.276·10-s − 0.642·11-s − 0.533·12-s − 1.72·13-s + 0.0863·14-s + 0.575i·15-s + 0.775·16-s − 0.610·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.849 - 0.527i$
Analytic conductor: \(27.7898\)
Root analytic conductor: \(5.27161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :3/2),\ -0.849 - 0.527i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3490373515\)
\(L(\frac12)\) \(\approx\) \(0.3490373515\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
157 \( 1 + (1.67e3 + 1.03e3i)T \)
good2 \( 1 - 0.783iT - 8T^{2} \)
5 \( 1 + 11.1iT - 125T^{2} \)
7 \( 1 + 5.77iT - 343T^{2} \)
11 \( 1 + 23.4T + 1.33e3T^{2} \)
13 \( 1 + 81.0T + 2.19e3T^{2} \)
17 \( 1 + 42.8T + 4.91e3T^{2} \)
19 \( 1 + 53.6T + 6.85e3T^{2} \)
23 \( 1 - 47.0iT - 1.21e4T^{2} \)
29 \( 1 - 312. iT - 2.43e4T^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 + 76.5T + 5.06e4T^{2} \)
41 \( 1 + 111. iT - 6.89e4T^{2} \)
43 \( 1 - 465. iT - 7.95e4T^{2} \)
47 \( 1 + 590.T + 1.03e5T^{2} \)
53 \( 1 - 462. iT - 1.48e5T^{2} \)
59 \( 1 - 165. iT - 2.05e5T^{2} \)
61 \( 1 + 712. iT - 2.26e5T^{2} \)
67 \( 1 + 546.T + 3.00e5T^{2} \)
71 \( 1 + 600.T + 3.57e5T^{2} \)
73 \( 1 + 995. iT - 3.89e5T^{2} \)
79 \( 1 - 174. iT - 4.93e5T^{2} \)
83 \( 1 + 748. iT - 5.71e5T^{2} \)
89 \( 1 + 727.T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3iT - 9.12e5T^{2} \)
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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

−10.91542114754199239393986023333, −10.27605944879443366184756520968, −9.187222457662363692939136680543, −8.056579806949101796900220808902, −7.25082984112245993999565859401, −6.40733420580714947360552275097, −5.18771779361240143044998190981, −4.67197023127818309627460621106, −2.82215188235509336862166630256, −1.48936752234332766590469264311, 0.10709028078117324735991670958, 2.21505798220686285383690429317, 2.78609795488270125040169040215, 4.40709480706203602655992681035, 5.65614496217326504758119217255, 6.65634104472884272469095184142, 7.18003150724242542910766116573, 8.239330510961201941325124724339, 9.928505293454138024875317079154, 10.24348218507158490798622125737

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