Properties

Label 2-930-1.1-c1-0-19
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 + 2·17-s − 18-s − 8·19-s + 20-s − 2·21-s + 4·22-s − 8·23-s − 24-s + 25-s + 4·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 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 + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.436·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-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: 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−9.857871322089170266577773826021, −8.786867640490033265070816077325, −8.097900309929688266359659768172, −7.27522620377953872578438371120, −6.37660797125702841019017938062, −5.42336948187457809337733615653, −4.11437368761771982050843164042, −2.76004182532151751361186145559, −2.10048574134409610178019006299, 0, 2.10048574134409610178019006299, 2.76004182532151751361186145559, 4.11437368761771982050843164042, 5.42336948187457809337733615653, 6.37660797125702841019017938062, 7.27522620377953872578438371120, 8.097900309929688266359659768172, 8.786867640490033265070816077325, 9.857871322089170266577773826021

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