Properties

Label 2-930-1.1-c1-0-18
Degree $2$
Conductor $930$
Sign $-1$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 2·7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 6·17-s − 18-s + 20-s − 2·21-s + 4·22-s + 24-s + 25-s + 4·26-s − 27-s + 2·28-s + 4·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.182·30-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: yes
Arithmetic: yes
Character: $\chi_{930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.938516725811639311622469454244, −8.751231633184441460023520408373, −8.042187355948291007372997726862, −7.14015673043828703735564682873, −6.35000905798728458255379424855, −5.18396576920330317497643634784, −4.64690014600703193412530855238, −2.76317445661407718659262659727, −1.78369370615482721230560496899, 0, 1.78369370615482721230560496899, 2.76317445661407718659262659727, 4.64690014600703193412530855238, 5.18396576920330317497643634784, 6.35000905798728458255379424855, 7.14015673043828703735564682873, 8.042187355948291007372997726862, 8.751231633184441460023520408373, 9.938516725811639311622469454244

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