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Properties

Label 2-1911-273.110-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.878 + 0.478i$
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  − i·3-s + (0.866 + 0.5i)4-s − 9-s + (0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + (0.366 + 0.366i)19-s + (0.866 − 0.5i)25-s + i·27-s + (0.366 − 1.36i)31-s + (−0.866 − 0.5i)36-s + (0.5 − 1.86i)37-s + (0.5 − 0.866i)39-s + (−1.5 + 0.866i)43-s + (0.866 − 0.499i)48-s + ⋯
L(s)  = 1  − i·3-s + (0.866 + 0.5i)4-s − 9-s + (0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + (0.366 + 0.366i)19-s + (0.866 − 0.5i)25-s + i·27-s + (0.366 − 1.36i)31-s + (−0.866 − 0.5i)36-s + (0.5 − 1.86i)37-s + (0.5 − 0.866i)39-s + (−1.5 + 0.866i)43-s + (0.866 − 0.499i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.878 + 0.478i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.878 + 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.437509767\)
\(L(\frac12)\) \(\approx\) \(1.437509767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.063131585887644502378490976070, −8.387599841706556734929025980542, −7.66320487917024068513435106993, −7.04282197979053829740615307553, −6.24475156984861357190844318170, −5.73588400531937188413042981378, −4.26581400444214462470810069566, −3.21853185773326238921527221135, −2.34683766658214896477436812572, −1.34123154517580344023955600913, 1.36552406150698187879630309829, 2.87668400731527366152353534366, 3.40677505305503168240067982979, 4.75978207064520881903208103721, 5.34713816254136412432785363583, 6.25722816806727583209750234206, 6.91090079219029806182638278918, 8.050110924670381259549895166466, 8.742130195514448119479012354653, 9.594882536988201810712002558015

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