Properties

Degree $2$
Conductor $930$
Sign $1$
Motivic weight $1$
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 + 4·7-s − 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s − 4·14-s − 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s + 4·21-s + 4·22-s + 4·23-s − 24-s + 25-s − 2·26-s + 27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.872·21-s + 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-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: $1$
Motivic weight: \(1\)
Character: $\chi_{930} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535481314\)
\(L(\frac12)\) \(\approx\) \(1.535481314\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.56889180562016, −18.84203919060033, −18.26475009399208, −17.78797523042747, −16.92190919274655, −15.98011452915491, −15.51923533043267, −14.72394872170646, −14.10926613558881, −13.20965089388011, −12.30321770126229, −11.36753491241698, −10.89472812141903, −10.07202789164518, −9.014193242451708, −8.337980306709956, −7.705769961375435, −7.201851756613001, −5.624141123660374, −4.832751545609514, −3.542239817942443, −2.396342519043293, −1.155844572250685, 1.155844572250685, 2.396342519043293, 3.542239817942443, 4.832751545609514, 5.624141123660374, 7.201851756613001, 7.705769961375435, 8.337980306709956, 9.014193242451708, 10.07202789164518, 10.89472812141903, 11.36753491241698, 12.30321770126229, 13.20965089388011, 14.10926613558881, 14.72394872170646, 15.51923533043267, 15.98011452915491, 16.92190919274655, 17.78797523042747, 18.26475009399208, 18.84203919060033, 19.56889180562016

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