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Properties

Label 2-930-5.4-c1-0-21
Degree $2$
Conductor $930$
Sign $0.165 + 0.986i$
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 − i·3-s − 4-s + (−0.369 − 2.20i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (2.20 − 0.369i)10-s + 3.26·11-s + i·12-s − 2.73i·13-s − 2·14-s + (−2.20 + 0.369i)15-s + 16-s − 4.93i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.165 − 0.986i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (0.697 − 0.116i)10-s + 0.983·11-s + 0.288i·12-s − 0.759i·13-s − 0.534·14-s + (−0.569 + 0.0954i)15-s + 0.250·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855363 - 0.723923i\)
\(L(\frac12)\) \(\approx\) \(0.855363 - 0.723923i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (0.369 + 2.20i)T \)
31 \( 1 + T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 3.26T + 11T^{2} \)
13 \( 1 + 2.73iT - 13T^{2} \)
17 \( 1 + 4.93iT - 17T^{2} \)
19 \( 1 + 4.93T + 19T^{2} \)
23 \( 1 - 0.521iT - 23T^{2} \)
29 \( 1 + 0.521T + 29T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8.82iT - 43T^{2} \)
47 \( 1 - 4.93iT - 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 + 9.34T + 59T^{2} \)
61 \( 1 + 9.75T + 61T^{2} \)
67 \( 1 + 13.5iT - 67T^{2} \)
71 \( 1 - 5.56T + 71T^{2} \)
73 \( 1 - 3.04iT - 73T^{2} \)
79 \( 1 - 6.41T + 79T^{2} \)
83 \( 1 - 1.36iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 7.26iT - 97T^{2} \)
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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

−9.278107629284690472144276525425, −9.086657328676041685426707427068, −8.153636414498025660664739553498, −7.41937040255015665357575594626, −6.40299417236218662198232875328, −5.61558989041235589979453314184, −4.82320216324444104996066113937, −3.66575857188705674443097216072, −2.09185019993051032073452075394, −0.53985296122424655334859698180, 1.63117750483127575910222366453, 2.99862139772311652847097024544, 4.02776596307716053345255355874, 4.39630658267190707424944343763, 6.09406640304293341632189353782, 6.70218853915673204038048057002, 7.85142825493578631508749918709, 8.797063629765069323012637022166, 9.641108318083680428172802930666, 10.46655825348802805256756301066

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