Properties

Label 2-345-1.1-c1-0-2
Degree $2$
Conductor $345$
Sign $1$
Analytic cond. $2.75483$
Root an. cond. $1.65977$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 3·8-s + 9-s + 10-s − 4·11-s − 12-s − 2·13-s − 4·14-s − 15-s − 16-s + 6·17-s − 18-s + 8·19-s + 20-s + 4·21-s + 4·22-s − 23-s + 3·24-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 + 1.06·8-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 + 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.872·21-s + 0.852·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(2.75483\)
Root analytic conductor: \(1.65977\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.060485995\)
\(L(\frac12)\) \(\approx\) \(1.060485995\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−11.45318892436668850123331148382, −10.22624874097662953151100700478, −9.780934304363534291352228156297, −8.370994697894994540574377339385, −7.988650960205812673256585775065, −7.41806121252840648222513928337, −5.23220582565731887389628167874, −4.65379839407946914161137175577, −3.04286276435946540203928099467, −1.26454930821134018164666796292, 1.26454930821134018164666796292, 3.04286276435946540203928099467, 4.65379839407946914161137175577, 5.23220582565731887389628167874, 7.41806121252840648222513928337, 7.988650960205812673256585775065, 8.370994697894994540574377339385, 9.780934304363534291352228156297, 10.22624874097662953151100700478, 11.45318892436668850123331148382

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