Properties

Degree 2
Conductor $ 3 \cdot 23 \cdot 29 $
Sign $-0.189 - 0.981i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + 3-s i·4-s + (−1 + i)6-s + 9-s i·12-s + 2i·13-s + 16-s + (−1 + i)18-s i·23-s + 25-s + (−2 − 2i)26-s + 27-s + i·29-s + (−1 − i)31-s + (−1 + i)32-s + ⋯
L(s)  = 1  + (−1 + i)2-s + 3-s i·4-s + (−1 + i)6-s + 9-s i·12-s + 2i·13-s + 16-s + (−1 + i)18-s i·23-s + 25-s + (−2 − 2i)26-s + 27-s + i·29-s + (−1 − i)31-s + (−1 + i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2001\)    =    \(3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-0.189 - 0.981i$
motivic weight  =  \(0\)
character  :  $\chi_{2001} (1172, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2001,\ (\ :0),\ -0.189 - 0.981i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9781764484\)
\(L(\frac12)\)  \(\approx\)  \(0.9781764484\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;23,\;29\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
23 \( 1 + iT \)
29 \( 1 - iT \)
good2 \( 1 + (1 - i)T - iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - iT^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.100987847803824006949055037011, −8.890001531197521807206290837304, −8.133882242002105114955438928583, −7.11460933333027645543764048647, −6.95618804459985683326613097557, −5.98407845037203523055299848919, −4.65406684250847171309839188439, −3.85595751568312228144040158792, −2.63311178906281489371066347836, −1.44824485471940837303575561174, 0.986810074274520295692088188142, 2.13755445319794556580613859946, 3.05820833930307492662366687208, 3.57422067079487566364897976209, 4.99828753289512800788046547803, 5.93340839207471549138623551794, 7.31186983166383381577428667384, 7.87582107007459813947343431808, 8.512134568620150899138281541910, 9.226196205317528357531239238560

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