Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.426 + 0.904i$
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 + (−0.953 + 2.02i)5-s − 5.11i·7-s − 9-s + 3.39·11-s − 5.34i·13-s + (−2.02 − 0.953i)15-s − 1.17i·17-s + 4.64·19-s + 5.11·21-s + 8.93i·23-s + (−3.17 − 3.85i)25-s i·27-s − 7.07·29-s − 9.13·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.426 + 0.904i)5-s − 1.93i·7-s − 0.333·9-s + 1.02·11-s − 1.48i·13-s + (−0.522 − 0.246i)15-s − 0.284i·17-s + 1.06·19-s + 1.11·21-s + 1.86i·23-s + (−0.635 − 0.771i)25-s − 0.192i·27-s − 1.31·29-s − 1.64·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.426 + 0.904i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.426 + 0.904i)$
$L(1)$  $\approx$  $0.8942795338$
$L(\frac12)$  $\approx$  $0.8942795338$
$L(\frac{3}{2})$   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 \( 1 + (0.953 - 2.02i)T \)
67 \( 1 - iT \)
good7 \( 1 + 5.11iT - 7T^{2} \)
11 \( 1 - 3.39T + 11T^{2} \)
13 \( 1 + 5.34iT - 13T^{2} \)
17 \( 1 + 1.17iT - 17T^{2} \)
19 \( 1 - 4.64T + 19T^{2} \)
23 \( 1 - 8.93iT - 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + 9.13T + 31T^{2} \)
37 \( 1 + 5.98iT - 37T^{2} \)
41 \( 1 - 6.63T + 41T^{2} \)
43 \( 1 + 7.21iT - 43T^{2} \)
47 \( 1 + 3.89iT - 47T^{2} \)
53 \( 1 - 5.25iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 0.377T + 61T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 0.211iT - 73T^{2} \)
79 \( 1 + 0.440T + 79T^{2} \)
83 \( 1 + 1.58iT - 83T^{2} \)
89 \( 1 - 7.78T + 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−7.79375236056003969528785455721, −7.48243791219186313882914171224, −7.07936500116283595897480353955, −5.91284016783729231852692162200, −5.25912068728688964309819990041, −3.95679621198802477585191041413, −3.74706555020959915314578118512, −3.09108409935776163624173420471, −1.45338630056634354206845248445, −0.26059526607886294606792086047, 1.42526615681896335652131050040, 2.05831674961837051763138483239, 3.15483248718756557746787018821, 4.24004774496795746199303830253, 4.95700366787589508926044762463, 5.86137348894524034689578251829, 6.32236502047386680838842526051, 7.24030077362879268951604048210, 8.088056512365733970109954053418, 8.827134667992159219550878658008

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