Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \cdot 7 $
Sign $0.100 - 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44i·3-s + (2.22 + 0.224i)5-s i·7-s − 2.99·9-s + 4.89·11-s + 4.44i·13-s + (−0.550 + 5.44i)15-s − 2i·17-s + 1.55·19-s + 2.44·21-s − 2.89i·23-s + (4.89 + i)25-s − 6.89·29-s − 8.89·31-s + 11.9i·33-s + ⋯
L(s)  = 1  + 1.41i·3-s + (0.994 + 0.100i)5-s − 0.377i·7-s − 0.999·9-s + 1.47·11-s + 1.23i·13-s + (−0.142 + 1.40i)15-s − 0.485i·17-s + 0.355·19-s + 0.534·21-s − 0.604i·23-s + (0.979 + 0.200i)25-s − 1.28·29-s − 1.59·31-s + 2.08i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.100 - 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{560} (449, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 560,\ (\ :1/2),\ 0.100 - 0.994i)$
$L(1)$  $\approx$  $1.33892 + 1.21048i$
$L(\frac12)$  $\approx$  $1.33892 + 1.21048i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.22 - 0.224i)T \)
7 \( 1 + iT \)
good3 \( 1 - 2.44iT - 3T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 4.44iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 1.55T + 19T^{2} \)
23 \( 1 + 2.89iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 + 8.89T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 - 0.898iT - 43T^{2} \)
47 \( 1 - 8.89iT - 47T^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 - 2.89iT - 73T^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 - 2.44iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.94804752328113704417831773016, −9.853759264125817213968967314347, −9.353093546516759192468927062867, −8.914828962261676706959211472869, −7.19258741600170091374277019536, −6.34011920926801764563699016675, −5.25152056206810604680905359809, −4.29419513522135571983399784796, −3.48738172840794368963940001970, −1.79534126096451592561920406277, 1.22009592651033190998187781891, 2.12770916693379347348022312685, 3.56234497325956004639105698721, 5.40937285065601616465906456078, 6.02786747564235330744470976082, 6.90815536370214047351725288857, 7.73995901752512594892029298067, 8.813383484243840707108756574536, 9.493179196316608173309848977406, 10.60519238475688134623755721642

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