Properties

Label 2-344760-1.1-c1-0-28
Degree $2$
Conductor $344760$
Sign $1$
Analytic cond. $2752.92$
Root an. cond. $52.4682$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 4·7-s + 9-s − 15-s + 17-s + 4·21-s + 25-s + 27-s − 2·29-s − 4·31-s − 4·35-s + 2·37-s + 6·41-s + 4·43-s − 45-s + 12·47-s + 9·49-s + 51-s + 6·53-s + 8·59-s + 14·61-s + 4·63-s − 8·67-s + 12·71-s − 2·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.258·15-s + 0.242·17-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.140·51-s + 0.824·53-s + 1.04·59-s + 1.79·61-s + 0.503·63-s − 0.977·67-s + 1.42·71-s − 0.234·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2752.92\)
Root analytic conductor: \(52.4682\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{344760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 344760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.982290128\)
\(L(\frac12)\) \(\approx\) \(4.982290128\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.52966396047958, −12.06087381991093, −11.66522246374294, −11.14103819169290, −10.85186388608980, −10.41698895773101, −9.730142134510767, −9.356830957702968, −8.717989407472071, −8.460782944253768, −8.022024993940810, −7.457452138750037, −7.275625050031075, −6.723354423408339, −5.781731131377816, −5.575773293022581, −4.994053506190782, −4.352579823032766, −4.049920629628723, −3.604438874314628, −2.761783445305000, −2.334476640421816, −1.807573242540632, −1.103029249545585, −0.6140005242999064, 0.6140005242999064, 1.103029249545585, 1.807573242540632, 2.334476640421816, 2.761783445305000, 3.604438874314628, 4.049920629628723, 4.352579823032766, 4.994053506190782, 5.575773293022581, 5.781731131377816, 6.723354423408339, 7.275625050031075, 7.457452138750037, 8.022024993940810, 8.460782944253768, 8.717989407472071, 9.356830957702968, 9.730142134510767, 10.41698895773101, 10.85186388608980, 11.14103819169290, 11.66522246374294, 12.06087381991093, 12.52966396047958

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