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Properties

Label 2-471-157.156-c3-0-15
Degree $2$
Conductor $471$
Sign $0.720 + 0.693i$
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  − 4.28i·2-s − 3·3-s − 10.4·4-s − 1.35i·5-s + 12.8i·6-s + 9.03i·7-s + 10.3i·8-s + 9·9-s − 5.79·10-s − 62.7·11-s + 31.2·12-s − 37.4·13-s + 38.7·14-s + 4.05i·15-s − 39.0·16-s − 19.5·17-s + ⋯
L(s)  = 1  − 1.51i·2-s − 0.577·3-s − 1.30·4-s − 0.120i·5-s + 0.875i·6-s + 0.487i·7-s + 0.455i·8-s + 0.333·9-s − 0.183·10-s − 1.71·11-s + 0.750·12-s − 0.798·13-s + 0.739·14-s + 0.0697i·15-s − 0.609·16-s − 0.279·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.720 + 0.693i$
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: \((2,\ 471,\ (\ :3/2),\ 0.720 + 0.693i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8990205574\)
\(L(\frac12)\) \(\approx\) \(0.8990205574\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.41e3 - 1.36e3i)T \)
good2 \( 1 + 4.28iT - 8T^{2} \)
5 \( 1 + 1.35iT - 125T^{2} \)
7 \( 1 - 9.03iT - 343T^{2} \)
11 \( 1 + 62.7T + 1.33e3T^{2} \)
13 \( 1 + 37.4T + 2.19e3T^{2} \)
17 \( 1 + 19.5T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 16.1iT - 1.21e4T^{2} \)
29 \( 1 - 166. iT - 2.43e4T^{2} \)
31 \( 1 - 265.T + 2.97e4T^{2} \)
37 \( 1 + 55.4T + 5.06e4T^{2} \)
41 \( 1 - 325. iT - 6.89e4T^{2} \)
43 \( 1 + 28.1iT - 7.95e4T^{2} \)
47 \( 1 + 40.9T + 1.03e5T^{2} \)
53 \( 1 + 634. iT - 1.48e5T^{2} \)
59 \( 1 - 567. iT - 2.05e5T^{2} \)
61 \( 1 + 726. iT - 2.26e5T^{2} \)
67 \( 1 - 771.T + 3.00e5T^{2} \)
71 \( 1 + 997.T + 3.57e5T^{2} \)
73 \( 1 - 861. iT - 3.89e5T^{2} \)
79 \( 1 + 721. iT - 4.93e5T^{2} \)
83 \( 1 - 34.9iT - 5.71e5T^{2} \)
89 \( 1 - 127.T + 7.04e5T^{2} \)
97 \( 1 - 1.84e3iT - 9.12e5T^{2} \)
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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

−10.48048676130861674972664215917, −10.06040669133210483144740946856, −9.071726144520064983329522671592, −7.939086237186480368576934647239, −6.78192871008052637830663950933, −5.26665911808007470857282147958, −4.77167223349185302641185095453, −3.15122566619245102192763036404, −2.37105870495213162707480466762, −0.883738868839727432515410300008, 0.40921006053593046052204857395, 2.68160599804108902936516919630, 4.53113180309041227014503751441, 5.23048729765635726821939547054, 6.04838245306164092558073926016, 7.23655508803057494265696695984, 7.53906779804881530962608688095, 8.567083447556311636738179032915, 9.818035524353115143368775417110, 10.53312155001171743965087503828

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