Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 3.61·7-s + 9-s − 3.95·11-s + 5.10·13-s − 15-s + 4.16·17-s + 2.79·19-s + 3.61·21-s + 3.13·23-s + 25-s + 27-s − 1.00·29-s + 2.03·31-s − 3.95·33-s − 3.61·35-s − 6.39·37-s + 5.10·39-s − 0.0315·41-s + 1.97·43-s − 45-s + 3.94·47-s + 6.03·49-s + 4.16·51-s − 2.86·53-s + 3.95·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.36·7-s + 0.333·9-s − 1.19·11-s + 1.41·13-s − 0.258·15-s + 1.01·17-s + 0.640·19-s + 0.787·21-s + 0.653·23-s + 0.200·25-s + 0.192·27-s − 0.186·29-s + 0.364·31-s − 0.687·33-s − 0.610·35-s − 1.05·37-s + 0.817·39-s − 0.00492·41-s + 0.301·43-s − 0.149·45-s + 0.576·47-s + 0.862·49-s + 0.583·51-s − 0.392·53-s + 0.532·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8040} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8040,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.175945766$
$L(\frac12)$  $\approx$  $3.175945766$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
67 \( 1 - T \)
good7 \( 1 - 3.61T + 7T^{2} \)
11 \( 1 + 3.95T + 11T^{2} \)
13 \( 1 - 5.10T + 13T^{2} \)
17 \( 1 - 4.16T + 17T^{2} \)
19 \( 1 - 2.79T + 19T^{2} \)
23 \( 1 - 3.13T + 23T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 - 2.03T + 31T^{2} \)
37 \( 1 + 6.39T + 37T^{2} \)
41 \( 1 + 0.0315T + 41T^{2} \)
43 \( 1 - 1.97T + 43T^{2} \)
47 \( 1 - 3.94T + 47T^{2} \)
53 \( 1 + 2.86T + 53T^{2} \)
59 \( 1 + 3.52T + 59T^{2} \)
61 \( 1 - 3.00T + 61T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + 1.02T + 83T^{2} \)
89 \( 1 - 3.28T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−7.936822532755427346330894023441, −7.44955522338675951050317351312, −6.56416673092618137146169606106, −5.43407989881119587217140327234, −5.17680896662814434068879624140, −4.20849156299778503152722493078, −3.48220474653029176536943060550, −2.75916564878224707979032789946, −1.70394726455677187179456561056, −0.924196896347061758655866552655, 0.924196896347061758655866552655, 1.70394726455677187179456561056, 2.75916564878224707979032789946, 3.48220474653029176536943060550, 4.20849156299778503152722493078, 5.17680896662814434068879624140, 5.43407989881119587217140327234, 6.56416673092618137146169606106, 7.44955522338675951050317351312, 7.936822532755427346330894023441

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