Properties

Label 2-930-93.68-c1-0-40
Degree $2$
Conductor $930$
Sign $-0.227 + 0.973i$
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  + i·2-s + (1.32 − 1.11i)3-s − 4-s + (−0.866 − 0.5i)5-s + (1.11 + 1.32i)6-s + (−0.581 − 1.00i)7-s i·8-s + (0.531 − 2.95i)9-s + (0.5 − 0.866i)10-s + (−0.0889 + 0.154i)11-s + (−1.32 + 1.11i)12-s + (−4.88 − 2.82i)13-s + (1.00 − 0.581i)14-s + (−1.70 + 0.297i)15-s + 16-s + (1.93 + 3.35i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.767 − 0.641i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (0.453 + 0.542i)6-s + (−0.219 − 0.380i)7-s − 0.353i·8-s + (0.177 − 0.984i)9-s + (0.158 − 0.273i)10-s + (−0.0268 + 0.0464i)11-s + (−0.383 + 0.320i)12-s + (−1.35 − 0.782i)13-s + (0.269 − 0.155i)14-s + (−0.440 + 0.0768i)15-s + 0.250·16-s + (0.470 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661746 - 0.833840i\)
\(L(\frac12)\) \(\approx\) \(0.661746 - 0.833840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.32 + 1.11i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-5.12 + 2.17i)T \)
good7 \( 1 + (0.581 + 1.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0889 - 0.154i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.88 + 2.82i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.93 - 3.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.57 + 4.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + 5.56T + 29T^{2} \)
37 \( 1 + (-0.782 + 0.451i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.48 + 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.735 + 0.424i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.11iT - 47T^{2} \)
53 \( 1 + (-6.25 + 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.27 + 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.00227iT - 61T^{2} \)
67 \( 1 + (3.20 - 5.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.9 + 6.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.32 - 5.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.52 - 1.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.95 + 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.20T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−9.707465628515860653044760601794, −8.714803701044268398410470480397, −7.990300424132929558648438231877, −7.42623930154292356946542694643, −6.64392452231276340085224242041, −5.61933113144350760605883473936, −4.43732628782084764427045962477, −3.50395502073068834569359128304, −2.23170571294628694475196892977, −0.43164215478030126803823173739, 2.03394938177777848278408941748, 2.87934836820207334892902212841, 3.92569138310353041955437366049, 4.66469102166451264452481418805, 5.76123666740691081832752877521, 7.19432365462212040525521734587, 7.966730950079320150651741866432, 8.830945916601743075342932977715, 9.691131222415995417128647452166, 10.07006474172406910562015899883

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