Properties

Label 2-280-1.1-c5-0-4
Degree $2$
Conductor $280$
Sign $1$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 30.3·3-s + 25·5-s + 49·7-s + 678.·9-s − 376.·11-s − 84.0·13-s − 758.·15-s + 1.19e3·17-s − 1.10e3·19-s − 1.48e3·21-s − 840.·23-s + 625·25-s − 1.32e4·27-s − 7.78e3·29-s − 1.03e3·31-s + 1.14e4·33-s + 1.22e3·35-s − 1.18e4·37-s + 2.55e3·39-s + 5.83e3·41-s + 2.00e4·43-s + 1.69e4·45-s + 1.86e3·47-s + 2.40e3·49-s − 3.62e4·51-s + 2.71e4·53-s − 9.41e3·55-s + ⋯
L(s)  = 1  − 1.94·3-s + 0.447·5-s + 0.377·7-s + 2.79·9-s − 0.938·11-s − 0.137·13-s − 0.870·15-s + 1.00·17-s − 0.704·19-s − 0.735·21-s − 0.331·23-s + 0.200·25-s − 3.48·27-s − 1.71·29-s − 0.194·31-s + 1.82·33-s + 0.169·35-s − 1.42·37-s + 0.268·39-s + 0.542·41-s + 1.65·43-s + 1.24·45-s + 0.123·47-s + 0.142·49-s − 1.95·51-s + 1.33·53-s − 0.419·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8777678843\)
\(L(\frac12)\) \(\approx\) \(0.8777678843\)
\(L(\frac{7}{2})\) 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 \( 1 - 25T \)
7 \( 1 - 49T \)
good3 \( 1 + 30.3T + 243T^{2} \)
11 \( 1 + 376.T + 1.61e5T^{2} \)
13 \( 1 + 84.0T + 3.71e5T^{2} \)
17 \( 1 - 1.19e3T + 1.41e6T^{2} \)
19 \( 1 + 1.10e3T + 2.47e6T^{2} \)
23 \( 1 + 840.T + 6.43e6T^{2} \)
29 \( 1 + 7.78e3T + 2.05e7T^{2} \)
31 \( 1 + 1.03e3T + 2.86e7T^{2} \)
37 \( 1 + 1.18e4T + 6.93e7T^{2} \)
41 \( 1 - 5.83e3T + 1.15e8T^{2} \)
43 \( 1 - 2.00e4T + 1.47e8T^{2} \)
47 \( 1 - 1.86e3T + 2.29e8T^{2} \)
53 \( 1 - 2.71e4T + 4.18e8T^{2} \)
59 \( 1 - 6.17e3T + 7.14e8T^{2} \)
61 \( 1 - 1.64e4T + 8.44e8T^{2} \)
67 \( 1 - 2.37e4T + 1.35e9T^{2} \)
71 \( 1 + 3.92e4T + 1.80e9T^{2} \)
73 \( 1 - 8.07e4T + 2.07e9T^{2} \)
79 \( 1 + 6.25e4T + 3.07e9T^{2} \)
83 \( 1 - 7.34e4T + 3.93e9T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + 9.17e4T + 8.58e9T^{2} \)
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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

−10.86605855632655408182959033780, −10.47032053697530355204583599439, −9.462851173032396697996607734833, −7.79931898064355654792444636782, −6.89739676810237211617953848876, −5.65634974429323574879703168117, −5.36099569442125102891702181573, −4.08335927893765529112401797346, −1.92760779923858033828989494429, −0.59153553102278329838432045121, 0.59153553102278329838432045121, 1.92760779923858033828989494429, 4.08335927893765529112401797346, 5.36099569442125102891702181573, 5.65634974429323574879703168117, 6.89739676810237211617953848876, 7.79931898064355654792444636782, 9.462851173032396697996607734833, 10.47032053697530355204583599439, 10.86605855632655408182959033780

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