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Properties

Label 2-471-157.156-c1-0-13
Degree $2$
Conductor $471$
Sign $0.705 - 0.708i$
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  + 1.76i·2-s − 3-s − 1.10·4-s − 1.19i·5-s − 1.76i·6-s − 3.36i·7-s + 1.58i·8-s + 9-s + 2.10·10-s − 3.41·11-s + 1.10·12-s + 5.92·13-s + 5.92·14-s + 1.19i·15-s − 4.98·16-s + 5.32·17-s + ⋯
L(s)  = 1  + 1.24i·2-s − 0.577·3-s − 0.550·4-s − 0.533i·5-s − 0.718i·6-s − 1.27i·7-s + 0.559i·8-s + 0.333·9-s + 0.664·10-s − 1.02·11-s + 0.318·12-s + 1.64·13-s + 1.58·14-s + 0.308i·15-s − 1.24·16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.705 - 0.708i$
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: \((2,\ 471,\ (\ :1/2),\ 0.705 - 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18269 + 0.491659i\)
\(L(\frac12)\) \(\approx\) \(1.18269 + 0.491659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (8.83 - 8.88i)T \)
good2 \( 1 - 1.76iT - 2T^{2} \)
5 \( 1 + 1.19iT - 5T^{2} \)
7 \( 1 + 3.36iT - 7T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 - 5.92T + 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 - 6.59T + 19T^{2} \)
23 \( 1 + 5.36iT - 23T^{2} \)
29 \( 1 - 4.69iT - 29T^{2} \)
31 \( 1 + 1.97T + 31T^{2} \)
37 \( 1 + 4.07T + 37T^{2} \)
41 \( 1 + 3.22iT - 41T^{2} \)
43 \( 1 + 2.11iT - 43T^{2} \)
47 \( 1 + 1.55T + 47T^{2} \)
53 \( 1 - 9.86iT - 53T^{2} \)
59 \( 1 + 3.30iT - 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 + 6.28T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 6.38iT - 73T^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 - 4.86iT - 83T^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 - 9.09iT - 97T^{2} \)
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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

−10.84749414557749043355163109342, −10.47755448518493269027943174696, −9.082120534621918905978997779790, −8.062743812650062658910577989807, −7.43961630628811802709628674579, −6.54266258113142270252795963416, −5.52696097256216424724191342133, −4.87860028733588989100011295011, −3.48191463592249892436002039291, −1.04425217118114040903960684196, 1.35785532192814274742242541782, 2.81572728838155023526898963923, 3.56460994557261967150853545027, 5.29936886986559250014767701621, 5.94668809255384510392842317454, 7.22388435069687042524358473567, 8.379272803885808806834580328871, 9.558474758384499606855718432777, 10.18594198929098582166004314811, 11.18795381628086994319944048281

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