Properties

Label 2-56e2-1.1-c1-0-44
Degree $2$
Conductor $3136$
Sign $-1$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
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  − 2.82·3-s + 1.41·5-s + 5.00·9-s − 4·11-s + 4.24·13-s − 4.00·15-s − 1.41·17-s + 2.82·19-s − 4·23-s − 2.99·25-s − 5.65·27-s − 8·29-s + 11.3·33-s + 8·37-s − 12·39-s + 7.07·41-s + 4·43-s + 7.07·45-s − 5.65·47-s + 4.00·51-s − 10·53-s − 5.65·55-s − 8.00·57-s + 14.1·59-s − 7.07·61-s + 6·65-s + 11.3·69-s + ⋯
L(s)  = 1  − 1.63·3-s + 0.632·5-s + 1.66·9-s − 1.20·11-s + 1.17·13-s − 1.03·15-s − 0.342·17-s + 0.648·19-s − 0.834·23-s − 0.599·25-s − 1.08·27-s − 1.48·29-s + 1.96·33-s + 1.31·37-s − 1.92·39-s + 1.10·41-s + 0.609·43-s + 1.05·45-s − 0.825·47-s + 0.560·51-s − 1.37·53-s − 0.762·55-s − 1.05·57-s + 1.84·59-s − 0.905·61-s + 0.744·65-s + 1.36·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2.82T + 3T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 - 1.41T + 97T^{2} \)
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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

−8.124939593837381014927572600669, −7.50207714511441993113073894176, −6.50375889240135103698234272055, −5.82824166734359957270460478545, −5.57843676375437461370432574756, −4.65272111652703885058738402262, −3.74163182406624173881593430028, −2.37242599754759520820039395423, −1.25766049639397829647372456169, 0, 1.25766049639397829647372456169, 2.37242599754759520820039395423, 3.74163182406624173881593430028, 4.65272111652703885058738402262, 5.57843676375437461370432574756, 5.82824166734359957270460478545, 6.50375889240135103698234272055, 7.50207714511441993113073894176, 8.124939593837381014927572600669

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