Properties

Label 2-2842-1.1-c1-0-86
Degree $2$
Conductor $2842$
Sign $-1$
Analytic cond. $22.6934$
Root an. cond. $4.76376$
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 + 2.35·3-s + 4-s − 0.662·5-s − 2.35·6-s − 8-s + 2.56·9-s + 0.662·10-s − 0.438·11-s + 2.35·12-s − 4.05·13-s − 1.56·15-s + 16-s − 4.34·17-s − 2.56·18-s + 7.36·19-s − 0.662·20-s + 0.438·22-s − 8.24·23-s − 2.35·24-s − 4.56·25-s + 4.05·26-s − 1.03·27-s + 29-s + 1.56·30-s − 4.05·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.36·3-s + 0.5·4-s − 0.296·5-s − 0.962·6-s − 0.353·8-s + 0.853·9-s + 0.209·10-s − 0.132·11-s + 0.680·12-s − 1.12·13-s − 0.403·15-s + 0.250·16-s − 1.05·17-s − 0.603·18-s + 1.68·19-s − 0.148·20-s + 0.0934·22-s − 1.71·23-s − 0.481·24-s − 0.912·25-s + 0.795·26-s − 0.198·27-s + 0.185·29-s + 0.285·30-s − 0.728·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2842\)    =    \(2 \cdot 7^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(22.6934\)
Root analytic conductor: \(4.76376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2842,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
29 \( 1 - T \)
good3 \( 1 - 2.35T + 3T^{2} \)
5 \( 1 + 0.662T + 5T^{2} \)
11 \( 1 + 0.438T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
19 \( 1 - 7.36T + 19T^{2} \)
23 \( 1 + 8.24T + 23T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 + 4.34T + 41T^{2} \)
43 \( 1 + 4.68T + 43T^{2} \)
47 \( 1 + 0.662T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 - 1.32T + 61T^{2} \)
67 \( 1 + 4.87T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 7.80T + 79T^{2} \)
83 \( 1 + 5.08T + 83T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 + 6.99T + 97T^{2} \)
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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

−8.327476867170845756138707782220, −7.77196301892895755262920019141, −7.37601463565760103654700338043, −6.36887193651266158540024653883, −5.29943555319643442347434210663, −4.21285481231937893915527427589, −3.34918860537743190165069392337, −2.50321110819973847478567922670, −1.76244192578826469777742681651, 0, 1.76244192578826469777742681651, 2.50321110819973847478567922670, 3.34918860537743190165069392337, 4.21285481231937893915527427589, 5.29943555319643442347434210663, 6.36887193651266158540024653883, 7.37601463565760103654700338043, 7.77196301892895755262920019141, 8.327476867170845756138707782220

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