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Properties

Label 2-930-93.26-c1-0-7
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: $\chi_{930} (491, \cdot )$
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

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

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