Properties

Label 2-257600-1.1-c1-0-119
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 + 4·11-s + 3·13-s − 17-s + 23-s + 4·31-s − 11·37-s − 10·41-s − 2·43-s − 11·47-s + 49-s − 53-s − 8·59-s + 8·61-s − 3·63-s + 4·71-s − 4·73-s + 4·77-s + 11·79-s + 9·81-s − 13·83-s + 89-s + 3·91-s − 7·97-s − 12·99-s + 101-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s + 1.20·11-s + 0.832·13-s − 0.242·17-s + 0.208·23-s + 0.718·31-s − 1.80·37-s − 1.56·41-s − 0.304·43-s − 1.60·47-s + 1/7·49-s − 0.137·53-s − 1.04·59-s + 1.02·61-s − 0.377·63-s + 0.474·71-s − 0.468·73-s + 0.455·77-s + 1.23·79-s + 81-s − 1.42·83-s + 0.105·89-s + 0.314·91-s − 0.710·97-s − 1.20·99-s + 0.0995·101-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.13361089625145, −12.50356326969170, −11.98706671498460, −11.65235545320724, −11.31599799574421, −10.85692331674081, −10.31731903261871, −9.811292261616964, −9.255565406910189, −8.755153402460350, −8.407958585839633, −8.188956771334209, −7.347617557537979, −6.798505988449944, −6.443636187138401, −6.024916529286399, −5.337726739470262, −4.959750802279539, −4.384893703737495, −3.641824267073030, −3.389903646314907, −2.801192298349212, −1.860447455229482, −1.609983513902030, −0.8039444628185411, 0, 0.8039444628185411, 1.609983513902030, 1.860447455229482, 2.801192298349212, 3.389903646314907, 3.641824267073030, 4.384893703737495, 4.959750802279539, 5.337726739470262, 6.024916529286399, 6.443636187138401, 6.798505988449944, 7.347617557537979, 8.188956771334209, 8.407958585839633, 8.755153402460350, 9.255565406910189, 9.811292261616964, 10.31731903261871, 10.85692331674081, 11.31599799574421, 11.65235545320724, 11.98706671498460, 12.50356326969170, 13.13361089625145

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