Properties

Label 2-10-1.1-c3-0-0
Degree $2$
Conductor $10$
Sign $1$
Analytic cond. $0.590019$
Root an. cond. $0.768127$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 8·3-s + 4·4-s + 5·5-s − 16·6-s − 4·7-s + 8·8-s + 37·9-s + 10·10-s + 12·11-s − 32·12-s − 58·13-s − 8·14-s − 40·15-s + 16·16-s + 66·17-s + 74·18-s − 100·19-s + 20·20-s + 32·21-s + 24·22-s + 132·23-s − 64·24-s + 25·25-s − 116·26-s − 80·27-s − 16·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.53·3-s + 1/2·4-s + 0.447·5-s − 1.08·6-s − 0.215·7-s + 0.353·8-s + 1.37·9-s + 0.316·10-s + 0.328·11-s − 0.769·12-s − 1.23·13-s − 0.152·14-s − 0.688·15-s + 1/4·16-s + 0.941·17-s + 0.968·18-s − 1.20·19-s + 0.223·20-s + 0.332·21-s + 0.232·22-s + 1.19·23-s − 0.544·24-s + 1/5·25-s − 0.874·26-s − 0.570·27-s − 0.107·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $1$
Analytic conductor: \(0.590019\)
Root analytic conductor: \(0.768127\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8824450529\)
\(L(\frac12)\) \(\approx\) \(0.8824450529\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
5 \( 1 - p T \)
good3 \( 1 + 8 T + p^{3} T^{2} \)
7 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 - 132 T + p^{3} T^{2} \)
29 \( 1 + 90 T + p^{3} T^{2} \)
31 \( 1 - 152 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 + 438 T + p^{3} T^{2} \)
43 \( 1 - 32 T + p^{3} T^{2} \)
47 \( 1 + 204 T + p^{3} T^{2} \)
53 \( 1 - 222 T + p^{3} T^{2} \)
59 \( 1 - 420 T + p^{3} T^{2} \)
61 \( 1 - 902 T + p^{3} T^{2} \)
67 \( 1 + 1024 T + p^{3} T^{2} \)
71 \( 1 - 432 T + p^{3} T^{2} \)
73 \( 1 - 362 T + p^{3} T^{2} \)
79 \( 1 + 160 T + p^{3} T^{2} \)
83 \( 1 - 72 T + p^{3} T^{2} \)
89 \( 1 - 810 T + p^{3} T^{2} \)
97 \( 1 - 1106 T + p^{3} T^{2} \)
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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

−21.13241426601042698123302530001, −19.13502572048254465535546332170, −17.34666090101985014439637973877, −16.62625077244728570466028148690, −14.82732212914541437300795209669, −12.88296977032189903354062156580, −11.73684702468117610360739302517, −10.21660783105547989838934329434, −6.67217788541912559504889095219, −5.12936292015576469502554864524, 5.12936292015576469502554864524, 6.67217788541912559504889095219, 10.21660783105547989838934329434, 11.73684702468117610360739302517, 12.88296977032189903354062156580, 14.82732212914541437300795209669, 16.62625077244728570466028148690, 17.34666090101985014439637973877, 19.13502572048254465535546332170, 21.13241426601042698123302530001

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