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 − 3.96·7-s + 8-s + 10-s − 3.04·11-s + 0.919·13-s − 3.96·14-s + 16-s − 0.351·17-s − 0.351·19-s + 20-s − 3.04·22-s + 3.43·23-s + 25-s + 0.919·26-s − 3.96·28-s − 9.37·29-s + 2.91·31-s + 32-s − 0.351·34-s − 3.96·35-s + 5.40·37-s − 0.351·38-s + 40-s + 8.45·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.50·7-s + 0.353·8-s + 0.316·10-s − 0.919·11-s + 0.254·13-s − 1.06·14-s + 0.250·16-s − 0.0853·17-s − 0.0807·19-s + 0.223·20-s − 0.650·22-s + 0.715·23-s + 0.200·25-s + 0.180·26-s − 0.750·28-s − 1.74·29-s + 0.524·31-s + 0.176·32-s − 0.0603·34-s − 0.670·35-s + 0.888·37-s − 0.0570·38-s + 0.158·40-s + 1.31·41-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  =  \((2,\ 6030,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.562739145\)
\(L(\frac12)\)  \(\approx\)  \(2.562739145\)
\(L(\frac{3}{2})\)   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 \( 1 + 3.96T + 7T^{2} \)
11 \( 1 + 3.04T + 11T^{2} \)
13 \( 1 - 0.919T + 13T^{2} \)
17 \( 1 + 0.351T + 17T^{2} \)
19 \( 1 + 0.351T + 19T^{2} \)
23 \( 1 - 3.43T + 23T^{2} \)
29 \( 1 + 9.37T + 29T^{2} \)
31 \( 1 - 2.91T + 31T^{2} \)
37 \( 1 - 5.40T + 37T^{2} \)
41 \( 1 - 8.45T + 41T^{2} \)
43 \( 1 - 6.09T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + 5.93T + 53T^{2} \)
59 \( 1 - 8.45T + 59T^{2} \)
61 \( 1 + 1.96T + 61T^{2} \)
71 \( 1 - 5.61T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + 0.105T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 3.04T + 97T^{2} \)
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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

−7.79196754163964758584457197370, −7.29792573233465699145582275364, −6.44937470681751327200263546527, −5.91063647297601553801023757949, −5.38299390647734274583868840495, −4.37912011577070927089298769787, −3.60768236894160735178602150946, −2.82408507323631665595631495747, −2.23041256326920294202043202255, −0.72780526784434483781654196629, 0.72780526784434483781654196629, 2.23041256326920294202043202255, 2.82408507323631665595631495747, 3.60768236894160735178602150946, 4.37912011577070927089298769787, 5.38299390647734274583868840495, 5.91063647297601553801023757949, 6.44937470681751327200263546527, 7.29792573233465699145582275364, 7.79196754163964758584457197370

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