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Properties

Label 2-471-157.144-c1-0-8
Degree $2$
Conductor $471$
Sign $0.753 - 0.657i$
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  − 0.907·2-s + (−0.5 + 0.866i)3-s − 1.17·4-s + (−0.0276 + 0.0478i)5-s + (0.453 − 0.785i)6-s + 1.18·7-s + 2.88·8-s + (−0.499 − 0.866i)9-s + (0.0250 − 0.0434i)10-s + (1.02 − 1.78i)11-s + (0.588 − 1.01i)12-s + (−1.49 − 2.58i)13-s − 1.07·14-s + (−0.0276 − 0.0478i)15-s − 0.263·16-s + (−1.22 + 2.12i)17-s + ⋯
L(s)  = 1  − 0.641·2-s + (−0.288 + 0.499i)3-s − 0.588·4-s + (−0.0123 + 0.0214i)5-s + (0.185 − 0.320i)6-s + 0.449·7-s + 1.01·8-s + (−0.166 − 0.288i)9-s + (0.00792 − 0.0137i)10-s + (0.310 − 0.537i)11-s + (0.169 − 0.294i)12-s + (−0.413 − 0.716i)13-s − 0.288·14-s + (−0.00713 − 0.0123i)15-s − 0.0658·16-s + (−0.297 + 0.516i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752831 + 0.282299i\)
\(L(\frac12)\) \(\approx\) \(0.752831 + 0.282299i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-12.5 - 0.0460i)T \)
good2 \( 1 + 0.907T + 2T^{2} \)
5 \( 1 + (0.0276 - 0.0478i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.18T + 7T^{2} \)
11 \( 1 + (-1.02 + 1.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.49 + 2.58i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.30 - 3.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.53T + 23T^{2} \)
29 \( 1 - 8.37T + 29T^{2} \)
31 \( 1 + (-2.32 - 4.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.50 - 6.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.54T + 41T^{2} \)
43 \( 1 + (4.70 + 8.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.23 - 9.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.611 + 1.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + (5.08 - 8.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 8.62T + 67T^{2} \)
71 \( 1 + (6.41 + 11.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.96 + 12.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.83T + 79T^{2} \)
83 \( 1 + (-2.53 - 4.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.02 + 6.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.93 - 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.64073054080353487854779830452, −10.44336463839217599810615466593, −9.223782909887832095491020574728, −8.598707898553718572310086810895, −7.78077403030153426975267874986, −6.51745794703475481701493572434, −5.24886422667713145853755770205, −4.51582660380309912488394964923, −3.21360665645768396350352833824, −1.11363266853748932937671969417, 0.865744180697272613433854281030, 2.42160298217550589040102403861, 4.45328126133395145197883613147, 4.95606624982090463520808655140, 6.60293252403889650979875682808, 7.26493665671246067826946635573, 8.356414963477146592092029676295, 9.046502921243768244347552840668, 9.897228734131252129595860926213, 10.92170886259116882195241025866

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