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Properties

Label 2-280-5.4-c5-0-25
Degree $2$
Conductor $280$
Sign $0.886 - 0.462i$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.80i·3-s + (49.5 − 25.8i)5-s + 49i·7-s + 209.·9-s + 415.·11-s + 419. i·13-s + (150. + 287. i)15-s − 47.1i·17-s − 1.30e3·19-s − 284.·21-s − 887. i·23-s + (1.78e3 − 2.56e3i)25-s + 2.62e3i·27-s + 7.31e3·29-s − 3.10e3·31-s + ⋯
L(s)  = 1  + 0.372i·3-s + (0.886 − 0.462i)5-s + 0.377i·7-s + 0.861·9-s + 1.03·11-s + 0.687i·13-s + (0.172 + 0.330i)15-s − 0.0395i·17-s − 0.826·19-s − 0.140·21-s − 0.349i·23-s + (0.571 − 0.820i)25-s + 0.693i·27-s + 1.61·29-s − 0.580·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.870163796\)
\(L(\frac12)\) \(\approx\) \(2.870163796\)
\(L(\frac{7}{2})\) 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 \( 1 + (-49.5 + 25.8i)T \)
7 \( 1 - 49iT \)
good3 \( 1 - 5.80iT - 243T^{2} \)
11 \( 1 - 415.T + 1.61e5T^{2} \)
13 \( 1 - 419. iT - 3.71e5T^{2} \)
17 \( 1 + 47.1iT - 1.41e6T^{2} \)
19 \( 1 + 1.30e3T + 2.47e6T^{2} \)
23 \( 1 + 887. iT - 6.43e6T^{2} \)
29 \( 1 - 7.31e3T + 2.05e7T^{2} \)
31 \( 1 + 3.10e3T + 2.86e7T^{2} \)
37 \( 1 + 3.81e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.04e3T + 1.15e8T^{2} \)
43 \( 1 + 8.66e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.23e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.23e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.19e4T + 7.14e8T^{2} \)
61 \( 1 - 1.64e4T + 8.44e8T^{2} \)
67 \( 1 - 2.89e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.52e4T + 1.80e9T^{2} \)
73 \( 1 - 5.47e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.79e4T + 3.07e9T^{2} \)
83 \( 1 - 9.32e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.27e5T + 5.58e9T^{2} \)
97 \( 1 + 1.02e5iT - 8.58e9T^{2} \)
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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

−10.95438278649767054783118859054, −9.982504984916510532236643004359, −9.249532870746788637008309551715, −8.527712088143126460131695568858, −6.93063254975603638553334191175, −6.15068423682976666834947306142, −4.86565320865004187328664696649, −4.00197255063097386221042787717, −2.25464864944665854140160837860, −1.14270660264733427328516025134, 0.967991456225453252359967568714, 2.04716972877557486087795037165, 3.51214327732507298112705328070, 4.81652196996012019725749927199, 6.25668338069727180844832540542, 6.79903317694295534640923467137, 7.925656067252891093892607397962, 9.167943077251667969031974896066, 10.08329148672805476246259132450, 10.73752137289725408668469795228

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