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Properties

Label 2-1911-273.227-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.101 + 0.994i$
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  + (0.866 − 0.5i)3-s − i·4-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s − 16-s + (0.5 − 0.133i)19-s + (−0.866 − 0.5i)25-s − 0.999i·27-s + (1.36 − 0.366i)31-s + (−0.866 − 0.499i)36-s + (1.36 + 1.36i)37-s − 0.999·39-s + (−1.5 + 0.866i)43-s + (−0.866 + 0.5i)48-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s − i·4-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)12-s + (−0.866 − 0.5i)13-s − 16-s + (0.5 − 0.133i)19-s + (−0.866 − 0.5i)25-s − 0.999i·27-s + (1.36 − 0.366i)31-s + (−0.866 − 0.499i)36-s + (1.36 + 1.36i)37-s − 0.999·39-s + (−1.5 + 0.866i)43-s + (−0.866 + 0.5i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.101 + 0.994i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.101 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.461619971\)
\(L(\frac12)\) \(\approx\) \(1.461619971\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
13 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.383131959065866767364196300334, −8.268111051787560474668964432536, −7.78411978418434573852284023834, −6.74823745791680558418774344140, −6.19328014374557248728714408148, −5.12464762132020746243492246045, −4.30864407201545803265209088050, −3.01561986832233034965255828174, −2.20846919428084148428309822645, −1.00795695278492357952105468651, 2.04362999192783597052779084478, 2.87402688673918490834133708238, 3.77486332705088833751454728420, 4.45851627802674402856486264615, 5.37906815928380402178022077558, 6.76461841569766865026191225314, 7.43920616960102607872641359283, 8.100963230336435847381519396050, 8.747981782693229332912467528086, 9.603494191013908134246046602095

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