Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.878 - 0.477i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (1.96 − 1.06i)5-s + 3.03i·7-s − 9-s + 2.01·11-s − 1.13i·13-s + (1.06 + 1.96i)15-s − 0.286i·17-s − 1.78·19-s − 3.03·21-s − 7.18i·23-s + (2.71 − 4.19i)25-s i·27-s + 8.82·29-s + 2.29·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.878 − 0.477i)5-s + 1.14i·7-s − 0.333·9-s + 0.607·11-s − 0.316i·13-s + (0.275 + 0.507i)15-s − 0.0693i·17-s − 0.408·19-s − 0.661·21-s − 1.49i·23-s + (0.543 − 0.839i)25-s − 0.192i·27-s + 1.63·29-s + 0.412·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.878 - 0.477i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.878 - 0.477i)$
$L(1)$  $\approx$  $2.433374591$
$L(\frac12)$  $\approx$  $2.433374591$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 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 \( 1 - iT \)
5 \( 1 + (-1.96 + 1.06i)T \)
67 \( 1 - iT \)
good7 \( 1 - 3.03iT - 7T^{2} \)
11 \( 1 - 2.01T + 11T^{2} \)
13 \( 1 + 1.13iT - 13T^{2} \)
17 \( 1 + 0.286iT - 17T^{2} \)
19 \( 1 + 1.78T + 19T^{2} \)
23 \( 1 + 7.18iT - 23T^{2} \)
29 \( 1 - 8.82T + 29T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 + 2.36iT - 37T^{2} \)
41 \( 1 - 5.15T + 41T^{2} \)
43 \( 1 - 2.11iT - 43T^{2} \)
47 \( 1 - 6.00iT - 47T^{2} \)
53 \( 1 + 9.22iT - 53T^{2} \)
59 \( 1 - 5.81T + 59T^{2} \)
61 \( 1 - 7.84T + 61T^{2} \)
71 \( 1 - 5.24T + 71T^{2} \)
73 \( 1 - 8.12iT - 73T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 - 9.42iT - 83T^{2} \)
89 \( 1 - 1.93T + 89T^{2} \)
97 \( 1 + 5.99iT - 97T^{2} \)
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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

−8.619424887308529275926685344234, −8.168082011326312123425483932918, −6.71828428885516372396512772101, −6.23154767815202346668974489965, −5.49376701073443031711272567402, −4.81752954096434130266807807799, −4.08829474368878240609888116350, −2.77740556660058258882459851399, −2.28816209694914649058411020391, −0.919922210727132227727114419108, 0.947537795720161870711994608971, 1.75701349095791856001780365091, 2.80128934230337973655240993982, 3.72340316240704040115128312261, 4.57347872191224689726967898128, 5.60383787719934154707189578710, 6.33746930304025812219124847784, 6.93084894898973541347335791485, 7.42113697170902796915811058131, 8.348443891474246056125442050329

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