Properties

Label 2-257600-1.1-c1-0-69
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  − 2·3-s + 7-s + 9-s + 4·11-s − 6·17-s − 6·19-s − 2·21-s − 23-s + 4·27-s − 10·29-s − 4·31-s − 8·33-s − 2·37-s − 10·41-s + 4·43-s + 12·47-s + 49-s + 12·51-s − 6·53-s + 12·57-s − 2·59-s + 63-s + 2·69-s + 8·71-s + 6·73-s + 4·77-s + 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.45·17-s − 1.37·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 1.39·33-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 1.58·57-s − 0.260·59-s + 0.125·63-s + 0.240·69-s + 0.949·71-s + 0.702·73-s + 0.455·77-s + 0.900·79-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: $\chi_{257600} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.86909063027807, −12.44797416402591, −12.17967465910501, −11.52759145584484, −11.21422060428546, −10.81210964982544, −10.65382547457438, −9.811203013343057, −9.316979611538319, −8.816641258132249, −8.599607675245902, −7.856363836943188, −7.233091586818986, −6.783415854660496, −6.375707626026878, −6.020122005151807, −5.401921631507721, −4.970906601689574, −4.372317384141384, −3.941234097159917, −3.522477577613317, −2.469043022828979, −1.975797101870310, −1.496441822572133, −0.5819593890292717, 0, 0.5819593890292717, 1.496441822572133, 1.975797101870310, 2.469043022828979, 3.522477577613317, 3.941234097159917, 4.372317384141384, 4.970906601689574, 5.401921631507721, 6.020122005151807, 6.375707626026878, 6.783415854660496, 7.233091586818986, 7.856363836943188, 8.599607675245902, 8.816641258132249, 9.316979611538319, 9.811203013343057, 10.65382547457438, 10.81210964982544, 11.21422060428546, 11.52759145584484, 12.17967465910501, 12.44797416402591, 12.86909063027807

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