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Properties

Label 2-3971-11.10-c0-0-3
Degree $2$
Conductor $3971$
Sign $-i$
Analytic cond. $1.98178$
Root an. cond. $1.40775$
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  + i·2-s − 3-s + 5-s − i·6-s + i·8-s + i·10-s − i·11-s − i·13-s − 15-s − 16-s + i·17-s + 22-s + 23-s − i·24-s + 26-s + 27-s + ⋯
L(s)  = 1  + i·2-s − 3-s + 5-s − i·6-s + i·8-s + i·10-s − i·11-s − i·13-s − 15-s − 16-s + i·17-s + 22-s + 23-s − i·24-s + 26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3971\)    =    \(11 \cdot 19^{2}\)
Sign: $-i$
Analytic conductor: \(1.98178\)
Root analytic conductor: \(1.40775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3971} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3971,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.206947764\)
\(L(\frac12)\) \(\approx\) \(1.206947764\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + iT \)
19 \( 1 \)
good2 \( 1 - iT - T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.587976973266440405048342280491, −8.063380729416151463749921005821, −7.11772732657241524658167043488, −6.32846827567612275585897265566, −5.94313355791055669638086974363, −5.45400127415466043604631165168, −4.84878791354176291758034908040, −3.36400564658332679429846420329, −2.45157352409287811534396133688, −1.12964529689802170558547733747, 0.899859455520425706678539712851, 2.09484497331639200670216562841, 2.51808584642070507872985967415, 3.82675205510710825332624572545, 4.69597887753054413011126754000, 5.44145748455455180740676683229, 6.19503697452162891616855693371, 6.91762591003107852309266776539, 7.37284147705225461020858598986, 8.919594192766627244859070042293

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