Properties

Label 2-257600-1.1-c1-0-109
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 − 6·11-s − 13-s + 3·17-s − 4·19-s + 2·21-s − 23-s − 4·27-s + 6·29-s − 8·31-s − 12·33-s + 11·37-s − 2·39-s − 8·43-s + 3·47-s + 49-s + 6·51-s + 9·53-s − 8·57-s − 8·61-s + 63-s − 8·67-s − 2·69-s + 6·71-s − 2·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 2.08·33-s + 1.80·37-s − 0.320·39-s − 1.21·43-s + 0.437·47-s + 1/7·49-s + 0.840·51-s + 1.23·53-s − 1.05·57-s − 1.02·61-s + 0.125·63-s − 0.977·67-s − 0.240·69-s + 0.712·71-s − 0.234·73-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.08178399811520, −12.74113824509217, −12.26627774006197, −11.67843966062374, −11.13338867145571, −10.66682019632247, −10.16208298769478, −9.959158566805569, −9.117519442275751, −8.943565786078503, −8.117097340537116, −8.095313729853776, −7.624354981411144, −7.161675194545270, −6.462165055720429, −5.722700414876851, −5.518386776189306, −4.742742904074514, −4.429364661492448, −3.657194951055140, −3.162330556332362, −2.643891042375542, −2.243113470299544, −1.739674601046517, −0.7728114533748104, 0, 0.7728114533748104, 1.739674601046517, 2.243113470299544, 2.643891042375542, 3.162330556332362, 3.657194951055140, 4.429364661492448, 4.742742904074514, 5.518386776189306, 5.722700414876851, 6.462165055720429, 7.161675194545270, 7.624354981411144, 8.095313729853776, 8.117097340537116, 8.943565786078503, 9.117519442275751, 9.959158566805569, 10.16208298769478, 10.66682019632247, 11.13338867145571, 11.67843966062374, 12.26627774006197, 12.74113824509217, 13.08178399811520

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