Properties

Label 2-257600-1.1-c1-0-130
Degree $2$
Conductor $257600$
Sign $-1$
Analytic cond. $2056.94$
Root an. cond. $45.3535$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·9-s + 2·13-s + 2·17-s + 4·19-s + 23-s + 2·29-s + 2·37-s + 2·41-s − 4·43-s + 12·47-s + 49-s + 2·53-s − 12·59-s + 6·61-s + 3·63-s + 4·67-s + 10·73-s + 4·79-s + 9·81-s + 12·83-s + 6·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s + 0.371·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s + 0.768·61-s + 0.377·63-s + 0.488·67-s + 1.17·73-s + 0.450·79-s + 81-s + 1.31·83-s + 0.635·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(257600\)    =    \(2^{6} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(2056.94\)
Root analytic conductor: \(45.3535\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 257600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.17962228065526, −12.43188506749466, −12.11521972771143, −11.79464618521540, −11.11825130750239, −10.82375736319892, −10.38609737739009, −9.655066773931984, −9.405706894732383, −8.891268660027269, −8.405481764353403, −7.894373913042009, −7.528184908951729, −6.848000214957108, −6.385756007546986, −5.914922675804694, −5.418615282732219, −5.047224996184980, −4.333171683499243, −3.572264592548029, −3.423392145635526, −2.624591762942028, −2.316061143978749, −1.273544330124549, −0.8492412130453077, 0, 0.8492412130453077, 1.273544330124549, 2.316061143978749, 2.624591762942028, 3.423392145635526, 3.572264592548029, 4.333171683499243, 5.047224996184980, 5.418615282732219, 5.914922675804694, 6.385756007546986, 6.848000214957108, 7.528184908951729, 7.894373913042009, 8.405481764353403, 8.891268660027269, 9.405706894732383, 9.655066773931984, 10.38609737739009, 10.82375736319892, 11.11825130750239, 11.79464618521540, 12.11521972771143, 12.43188506749466, 13.17962228065526

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