Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \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  + 2-s + 4-s + 5-s − 1.45·7-s + 8-s + 10-s + 3.50·11-s + 4.95·13-s − 1.45·14-s + 16-s + 3.19·17-s + 3.19·19-s + 20-s + 3.50·22-s − 4.15·23-s + 25-s + 4.95·26-s − 1.45·28-s + 3.25·29-s + 6.95·31-s + 32-s + 3.19·34-s − 1.45·35-s − 4.70·37-s + 3.19·38-s + 40-s − 8.21·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.548·7-s + 0.353·8-s + 0.316·10-s + 1.05·11-s + 1.37·13-s − 0.387·14-s + 0.250·16-s + 0.774·17-s + 0.732·19-s + 0.223·20-s + 0.748·22-s − 0.866·23-s + 0.200·25-s + 0.972·26-s − 0.274·28-s + 0.603·29-s + 1.25·31-s + 0.176·32-s + 0.547·34-s − 0.245·35-s − 0.773·37-s + 0.518·38-s + 0.158·40-s − 1.28·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6030,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.088038693\)
\(L(\frac12)\)  \(\approx\)  \(4.088038693\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
67 \( 1 + T \)
good7 \( 1 + 1.45T + 7T^{2} \)
11 \( 1 - 3.50T + 11T^{2} \)
13 \( 1 - 4.95T + 13T^{2} \)
17 \( 1 - 3.19T + 17T^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + 4.15T + 23T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 - 6.95T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 + 8.21T + 41T^{2} \)
43 \( 1 + 7.01T + 43T^{2} \)
47 \( 1 - 8.66T + 47T^{2} \)
53 \( 1 + 0.900T + 53T^{2} \)
59 \( 1 + 8.21T + 59T^{2} \)
61 \( 1 - 0.549T + 61T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 - 7.40T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 3.50T + 97T^{2} \)
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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

−8.137607648416466910642709241274, −7.09446130712988354304759811238, −6.43684758541010429171879363814, −6.05310109088027392232128140833, −5.27855666441404565704171590217, −4.38168343531070270312768135012, −3.52598702621087233571197751013, −3.12906884707229505517340483580, −1.83442378977609818111514535246, −1.03682371434504502760165639563, 1.03682371434504502760165639563, 1.83442378977609818111514535246, 3.12906884707229505517340483580, 3.52598702621087233571197751013, 4.38168343531070270312768135012, 5.27855666441404565704171590217, 6.05310109088027392232128140833, 6.43684758541010429171879363814, 7.09446130712988354304759811238, 8.137607648416466910642709241274

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