Properties

Label 2-930-31.25-c1-0-9
Degree $2$
Conductor $930$
Sign $0.275 - 0.961i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (1.5 + 2.59i)7-s + 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−2.5 + 4.33i)11-s + (0.5 + 0.866i)12-s + (−3 + 5.19i)13-s + (1.5 + 2.59i)14-s + 0.999·15-s + 16-s + (−4 − 6.92i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (0.566 + 0.981i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.753 + 1.30i)11-s + (0.144 + 0.249i)12-s + (−0.832 + 1.44i)13-s + (0.400 + 0.694i)14-s + 0.258·15-s + 0.250·16-s + (−0.970 − 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14213 + 1.61522i\)
\(L(\frac12)\) \(\approx\) \(2.14213 + 1.61522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-3.5 + 4.33i)T \)
good7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + (2.5 - 4.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−10.12007160013701459727914195549, −9.347403971485160717193803877670, −8.824277912017998811290746832557, −7.50807883207395849046759655625, −6.93026350648617093171363735046, −5.46959263109360948546350358763, −4.89841119727973533320497935899, −4.32394364756848824847662722274, −2.63069543971932632781557599185, −2.08174974302417422411168557278, 0.996437145237133674360229016805, 2.62841328024383300917228078445, 3.30873572242856530579676389920, 4.61181462844985176584149588873, 5.52674926280873650344152585339, 6.44200231245006661185708823773, 7.36984083209812424480092532595, 7.979285671058630070901909676871, 8.844716599488246756391889802239, 10.30478829125647750313917427193

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