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Properties

Label 2-1911-273.116-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.0725 + 0.997i$
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  + (0.923 + 1.60i)2-s + (−0.5 + 0.866i)3-s + (−1.20 + 2.09i)4-s + (−0.382 − 0.662i)5-s − 1.84·6-s − 2.61·8-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (−0.923 + 1.60i)11-s + (−1.20 − 2.09i)12-s − 13-s + 0.765·15-s + (−1.20 − 2.09i)16-s + (0.923 − 1.60i)18-s + 1.84·20-s + ⋯
L(s)  = 1  + (0.923 + 1.60i)2-s + (−0.5 + 0.866i)3-s + (−1.20 + 2.09i)4-s + (−0.382 − 0.662i)5-s − 1.84·6-s − 2.61·8-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (−0.923 + 1.60i)11-s + (−1.20 − 2.09i)12-s − 13-s + 0.765·15-s + (−1.20 − 2.09i)16-s + (0.923 − 1.60i)18-s + 1.84·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.0725 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7262191189\)
\(L(\frac12)\) \(\approx\) \(0.7262191189\)
\(L(1)\) 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 \( 1 + T \)
good2 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 0.765T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.84T + T^{2} \)
89 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.784485445314712216574268325553, −9.036196376149104388856881060965, −8.175895039941214735397707926112, −7.45262961193907714147007983993, −6.81612629083754089257562773023, −5.80009376091380386345730549083, −5.00494991879379168204540787217, −4.67664941923511768910989725927, −4.02229305405552366948908190894, −2.74217277227607786957084605597, 0.39843003109222045745526984364, 1.86739056322196572454119861399, 2.88020834224638559569301326556, 3.33089529554139247745752495567, 4.68555895387324774132927498577, 5.41562740817610054180924421561, 6.07693649734975772274156152669, 7.11074225309354953861947246672, 8.010952205100525363428193555048, 8.947458819005253849877695382586

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