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Properties

Label 2-1911-273.263-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.794 + 0.606i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
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  − 3-s + (−0.5 − 0.866i)4-s + 9-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (−0.5 + 0.866i)25-s − 27-s + (1 − 1.73i)31-s + (−0.5 − 0.866i)36-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + (0.5 − 0.866i)43-s + (0.499 − 0.866i)48-s + ⋯
L(s)  = 1  − 3-s + (−0.5 − 0.866i)4-s + 9-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + 19-s + (−0.5 + 0.866i)25-s − 27-s + (1 − 1.73i)31-s + (−0.5 − 0.866i)36-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + (0.5 − 0.866i)43-s + (0.499 − 0.866i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7545894737\)
\(L(\frac12)\) \(\approx\) \(0.7545894737\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.662533519613406673025375450039, −8.712823375299719566647542485077, −7.59237135916246935726521461125, −6.78439202537346547533396074802, −5.91798277993836865558801516530, −5.47902718184964787292789826781, −4.49543654635284036209384951782, −3.82984864835340156491718341275, −2.02927544728911295966570363215, −0.885188284368538904412591531769, 1.02725645095688147557234173972, 2.81096549727465753323732009727, 3.75096762742559148294459964529, 4.68075724484927963861264744053, 5.36045434435459453431243724307, 6.30802812004285915563331866730, 7.10471513391789749006308749361, 7.967863811140132526758333396627, 8.519721967645849578794199676846, 9.656256200553710211661201314125

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