Properties

Label 2-3234-1.1-c1-0-27
Degree $2$
Conductor $3234$
Sign $-1$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 3.46·5-s + 6-s − 8-s + 9-s + 3.46·10-s + 11-s − 12-s − 2·13-s + 3.46·15-s + 16-s + 3.46·17-s − 18-s + 1.46·19-s − 3.46·20-s − 22-s − 6.92·23-s + 24-s + 6.99·25-s + 2·26-s − 27-s − 6·29-s − 3.46·30-s + 1.46·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.54·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.894·15-s + 0.250·16-s + 0.840·17-s − 0.235·18-s + 0.335·19-s − 0.774·20-s − 0.213·22-s − 1.44·23-s + 0.204·24-s + 1.39·25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.632·30-s + 0.262·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 3.46T + 5T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 2.92T + 43T^{2} \)
47 \( 1 + 2.53T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 1.07T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−7.980839856472816231512541391835, −7.73075203722081899979590398312, −7.04724458323073659643565916361, −6.12233094968129953908267084798, −5.30632100621679819804951013256, −4.20554096578801022587513917028, −3.68522956008377636478202976477, −2.47323405734273934052503713735, −1.04983271370502453256593629120, 0, 1.04983271370502453256593629120, 2.47323405734273934052503713735, 3.68522956008377636478202976477, 4.20554096578801022587513917028, 5.30632100621679819804951013256, 6.12233094968129953908267084798, 7.04724458323073659643565916361, 7.73075203722081899979590398312, 7.980839856472816231512541391835

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