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Properties

Label 2-471-157.12-c1-0-8
Degree $2$
Conductor $471$
Sign $0.590 - 0.807i$
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  − 2.26·2-s + (−0.5 − 0.866i)3-s + 3.11·4-s + (1.51 + 2.61i)5-s + (1.13 + 1.95i)6-s + 4.51·7-s − 2.52·8-s + (−0.499 + 0.866i)9-s + (−3.41 − 5.92i)10-s + (2.39 + 4.14i)11-s + (−1.55 − 2.69i)12-s + (−0.539 + 0.933i)13-s − 10.2·14-s + (1.51 − 2.61i)15-s − 0.528·16-s + (0.551 + 0.955i)17-s + ⋯
L(s)  = 1  − 1.59·2-s + (−0.288 − 0.499i)3-s + 1.55·4-s + (0.675 + 1.17i)5-s + (0.461 + 0.799i)6-s + 1.70·7-s − 0.891·8-s + (−0.166 + 0.288i)9-s + (−1.08 − 1.87i)10-s + (0.721 + 1.24i)11-s + (−0.449 − 0.778i)12-s + (−0.149 + 0.259i)13-s − 2.72·14-s + (0.390 − 0.675i)15-s − 0.132·16-s + (0.133 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.590 - 0.807i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ 0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697194 + 0.353862i\)
\(L(\frac12)\) \(\approx\) \(0.697194 + 0.353862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
157 \( 1 + (1.12 + 12.4i)T \)
good2 \( 1 + 2.26T + 2T^{2} \)
5 \( 1 + (-1.51 - 2.61i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.51T + 7T^{2} \)
11 \( 1 + (-2.39 - 4.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.539 - 0.933i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.551 - 0.955i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.576 + 0.998i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 9.14T + 23T^{2} \)
29 \( 1 - 0.0486T + 29T^{2} \)
31 \( 1 + (3.69 - 6.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.96 + 8.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 + (2.25 - 3.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.536 + 0.929i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.443 - 0.768i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.89T + 59T^{2} \)
61 \( 1 + (0.325 + 0.564i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 7.64T + 67T^{2} \)
71 \( 1 + (-1.13 + 1.96i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.26 - 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.49T + 79T^{2} \)
83 \( 1 + (-6.07 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.58 + 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.88 + 4.99i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96263355616615321374725988324, −10.23480207942809696795704582436, −9.462600906903994710802552216621, −8.375849026830959319919550949050, −7.53191212956992337471266756462, −6.98063390731375440134958082134, −5.96239853593995929983692986224, −4.46230333872997012056801639211, −2.15905841499454228502450480361, −1.67896515823058870119823086218, 0.911567610894197623875489500884, 1.94379928084804447461003635084, 4.22232966075093041714238134648, 5.31240208726059647477142429326, 6.20705309302564494036969064478, 7.922710372072104955365188346156, 8.255559157296512085608366149624, 9.120249870271509059340031556945, 9.764151996958104232381987901198, 10.74734944524319637716689886010

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