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Properties

Label 2-930-31.16-c1-0-9
Degree $2$
Conductor $930$
Sign $0.681 - 0.732i$
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 + (−0.785 + 2.41i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.903 + 2.77i)11-s + (0.809 − 0.587i)12-s + (4.17 + 3.03i)13-s + (0.785 + 2.41i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.132 + 0.408i)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 + (−0.296 + 0.913i)7-s + (−0.109 − 0.336i)8-s + (0.103 + 0.317i)9-s + (−0.255 + 0.185i)10-s + (−0.272 + 0.838i)11-s + (0.233 − 0.169i)12-s + (1.15 + 0.840i)13-s + (0.209 + 0.645i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.0322 + 0.0991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.681 - 0.732i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.681 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00340 + 0.872530i\)
\(L(\frac12)\) \(\approx\) \(2.00340 + 0.872530i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.753 - 5.51i)T \)
good7 \( 1 + (0.785 - 2.41i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.903 - 2.77i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-4.17 - 3.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.132 - 0.408i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.30 - 0.945i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.461 - 1.41i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.21 + 0.882i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 6.99T + 37T^{2} \)
41 \( 1 + (-2.47 + 1.80i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-5.25 + 3.81i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (9.24 + 6.71i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.63 - 5.01i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.59 - 4.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 4.69T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 + (4.12 + 12.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.96 + 9.13i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.00 - 6.17i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.41 + 1.03i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.26 - 6.97i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.229 - 0.707i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.26714963748211525875624915991, −9.269063678037822050925100419775, −8.759280763403088210539930221351, −7.69924657814779156352807419725, −6.62448844095879478931415554374, −5.73291121072629010224882244120, −4.64835358121052252448569912010, −3.86219624568893057190449592889, −2.85697950473515874461088092168, −1.76267200575320768925863853036, 0.845790745236902371364267463352, 2.81292820577386048206759645697, 3.61800804309076824478383671222, 4.45288399493318207994271891219, 5.80930560844422695314393394085, 6.48191732024321752552447073608, 7.51135300353910396811239748492, 8.077312618788565872495869941130, 8.812244668135577605491595441516, 9.994935833298508702306425150712

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