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Properties

Label 2-1911-39.23-c0-0-1
Degree $2$
Conductor $1911$
Sign $-0.872 + 0.488i$
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  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + (−1.5 − 0.866i)19-s − 25-s + 0.999·27-s + (0.499 + 0.866i)36-s + (1.5 − 0.866i)37-s + (−0.499 + 0.866i)39-s + (0.5 − 0.866i)43-s + (−0.499 + 0.866i)48-s − 0.999·52-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s − 0.999·12-s + (−0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + (−1.5 − 0.866i)19-s − 25-s + 0.999·27-s + (0.499 + 0.866i)36-s + (1.5 − 0.866i)37-s + (−0.499 + 0.866i)39-s + (0.5 − 0.866i)43-s + (−0.499 + 0.866i)48-s − 0.999·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.872 + 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8203004147\)
\(L(\frac12)\) \(\approx\) \(0.8203004147\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.078603658779419018443433569790, −8.097128653855452684735595781446, −7.36350503116856883740770103992, −6.64516734715479297145623861589, −5.91877674034631193595653990602, −5.32468806028025142523235153501, −4.32451662173143775521012180716, −2.66745555936155289403605490349, −2.00166582799035419287843852625, −0.59619517928034155510826749509, 2.00500247912293567011383727501, 3.09553092393455181817991695072, 4.14964012363205981019583132946, 4.51609018354352792829453870868, 5.92768647642378177697410623732, 6.39896875523237896851727197586, 7.38330118863900126867678063803, 8.226722828839490055123986141565, 8.949804852530515720052663925626, 9.797350943505261902503758501593

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