Properties

Label 2-257400-1.1-c1-0-92
Degree $2$
Conductor $257400$
Sign $-1$
Analytic cond. $2055.34$
Root an. cond. $45.3359$
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  + 4·7-s + 11-s − 13-s − 2·17-s + 4·19-s − 4·23-s − 6·29-s − 10·37-s + 10·41-s − 4·43-s + 9·49-s + 10·53-s + 12·59-s − 2·61-s − 8·67-s + 8·71-s + 2·73-s + 4·77-s − 16·79-s − 12·83-s + 10·89-s − 4·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 1.11·29-s − 1.64·37-s + 1.56·41-s − 0.609·43-s + 9/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s + 0.949·71-s + 0.234·73-s + 0.455·77-s − 1.80·79-s − 1.31·83-s + 1.05·89-s − 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 257400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(257400\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(2055.34\)
Root analytic conductor: \(45.3359\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{257400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 257400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.05531582092961, −12.60023843254225, −11.89426594265630, −11.66853129534234, −11.37054504082071, −10.81658403384214, −10.23645869469460, −9.967389935798124, −9.232685306122691, −8.753490017718752, −8.530965666765850, −7.802825493806735, −7.399495878551196, −7.189738208863052, −6.383881303409351, −5.801433451023310, −5.377396433572393, −4.902157648723710, −4.424403947060865, −3.818946740362462, −3.429831260424610, −2.450951528492047, −2.102291099566826, −1.499990420984328, −0.9085784981482994, 0, 0.9085784981482994, 1.499990420984328, 2.102291099566826, 2.450951528492047, 3.429831260424610, 3.818946740362462, 4.424403947060865, 4.902157648723710, 5.377396433572393, 5.801433451023310, 6.383881303409351, 7.189738208863052, 7.399495878551196, 7.802825493806735, 8.530965666765850, 8.753490017718752, 9.232685306122691, 9.967389935798124, 10.23645869469460, 10.81658403384214, 11.37054504082071, 11.66853129534234, 11.89426594265630, 12.60023843254225, 13.05531582092961

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