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Properties

Label 2-2001-2001.1931-c0-0-4
Degree $2$
Conductor $2001$
Sign $-0.189 + 0.981i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9781764484\)
\(L(\frac12)\) \(\approx\) \(0.9781764484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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