Properties

Label 2-930-31.2-c1-0-8
Degree $2$
Conductor $930$
Sign $-0.292 - 0.956i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s − 6-s + (0.726 + 2.23i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (1.22 + 3.77i)11-s + (−0.809 − 0.587i)12-s + (3.41 − 2.47i)13-s + (−0.726 + 2.23i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.923 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.154 + 0.475i)4-s + 0.447·5-s − 0.408·6-s + (0.274 + 0.844i)7-s + (−0.109 + 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (0.369 + 1.13i)11-s + (−0.233 − 0.169i)12-s + (0.946 − 0.687i)13-s + (−0.194 + 0.597i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.224 − 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23596 + 1.67085i\)
\(L(\frac12)\) \(\approx\) \(1.23596 + 1.67085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 - T \)
31 \( 1 + (1.73 - 5.29i)T \)
good7 \( 1 + (-0.726 - 2.23i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.22 - 3.77i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.41 + 2.47i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.923 + 2.84i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.20 + 0.874i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.98 - 6.10i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.49 + 1.08i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 7.05T + 37T^{2} \)
41 \( 1 + (-2.07 - 1.51i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.23 - 2.35i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.92 + 1.40i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.37 - 10.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.3 + 7.50i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 0.0596T + 61T^{2} \)
67 \( 1 + 7.69T + 67T^{2} \)
71 \( 1 + (0.930 - 2.86i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.45 - 4.46i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.72 + 14.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.54 + 4.02i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.23 - 6.86i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.15 + 12.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.33129793335297571537119056196, −9.425926556178005656905893100695, −8.726213960567237894488228686042, −7.61494367735539654651422251588, −6.74453659775769227810628362723, −5.74280073425020109796601501974, −5.29154489484784580461446122514, −4.26470400902510582837731758570, −3.12105424757713854568754920758, −1.74362500334167859758140651767, 0.924818177445365202241718582464, 2.05098343162687327008699626022, 3.62397084631099440837486313823, 4.29745018075793590707551580666, 5.59510515572839201218530247917, 6.21565424050240221060791022442, 6.96603483113147910165355958131, 8.197115992343538830989679789963, 8.976687704016439532984034089251, 10.19523815979195901889183393595

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