Properties

Label 2-930-1.1-c1-0-9
Degree $2$
Conductor $930$
Sign $1$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 2.56·7-s + 8-s + 9-s + 10-s + 2.56·11-s − 12-s + 2·13-s + 2.56·14-s − 15-s + 16-s − 3.12·17-s + 18-s − 7.68·19-s + 20-s − 2.56·21-s + 2.56·22-s + 1.43·23-s − 24-s + 25-s + 2·26-s − 27-s + 2.56·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.968·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.772·11-s − 0.288·12-s + 0.554·13-s + 0.684·14-s − 0.258·15-s + 0.250·16-s − 0.757·17-s + 0.235·18-s − 1.76·19-s + 0.223·20-s − 0.558·21-s + 0.546·22-s + 0.299·23-s − 0.204·24-s + 0.200·25-s + 0.392·26-s − 0.192·27-s + 0.484·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.531019911\)
\(L(\frac12)\) \(\approx\) \(2.531019911\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
31 \( 1 - T \)
good7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + 7.68T + 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 7.12T + 29T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 - 7.43T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 + 7.68T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 4.31T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.58369742668169622881392498137, −9.135189465923416730781059868944, −8.454113429484776346681778764311, −7.27109803947625248603649878617, −6.36442623568392223854921019561, −5.82657404658869598588534061457, −4.59322043530120736241495437638, −4.18573751729956530611062536605, −2.50779933605166344714653458320, −1.35636171358021831321444547883, 1.35636171358021831321444547883, 2.50779933605166344714653458320, 4.18573751729956530611062536605, 4.59322043530120736241495437638, 5.82657404658869598588534061457, 6.36442623568392223854921019561, 7.27109803947625248603649878617, 8.454113429484776346681778764311, 9.135189465923416730781059868944, 10.58369742668169622881392498137

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