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Properties

Label 2-1911-1911.926-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.171 - 0.985i$
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  + (0.0747 − 0.997i)3-s + (−0.222 + 0.974i)4-s + (−0.988 + 0.149i)7-s + (−0.988 − 0.149i)9-s + (0.955 + 0.294i)12-s + (−0.733 + 0.680i)13-s + (−0.900 − 0.433i)16-s + (0.988 + 1.71i)19-s + (0.0747 + 0.997i)21-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (0.0747 − 0.997i)28-s + (0.222 + 0.385i)31-s + (0.365 − 0.930i)36-s + (0.326 + 1.42i)37-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)3-s + (−0.222 + 0.974i)4-s + (−0.988 + 0.149i)7-s + (−0.988 − 0.149i)9-s + (0.955 + 0.294i)12-s + (−0.733 + 0.680i)13-s + (−0.900 − 0.433i)16-s + (0.988 + 1.71i)19-s + (0.0747 + 0.997i)21-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (0.0747 − 0.997i)28-s + (0.222 + 0.385i)31-s + (0.365 − 0.930i)36-s + (0.326 + 1.42i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.171 - 0.985i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (926, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.171 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5645603382\)
\(L(\frac12)\) \(\approx\) \(0.5645603382\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0747 + 0.997i)T \)
7 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (0.733 - 0.680i)T \)
good2 \( 1 + (0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.988 + 0.149i)T^{2} \)
11 \( 1 + (-0.955 + 0.294i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (-0.0747 + 0.997i)T^{2} \)
31 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.326 - 1.42i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.365 - 0.930i)T^{2} \)
43 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
47 \( 1 + (-0.955 + 0.294i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.0747 - 0.997i)T^{2} \)
73 \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \)
79 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.547759689401847585111954588801, −8.678171450342486224980992542142, −7.916565088497816343608167273795, −7.38615221155389170844360376271, −6.55420115899457204630164233999, −5.89058025468820159948252556272, −4.68579928247951944018262275996, −3.49511648875964841170306413221, −2.92662348577323155168725202821, −1.72484319616741306799445689553, 0.39062590673733079100725066893, 2.39785096910024667243561079803, 3.32289593140366576662621928909, 4.33378824393841081868636060129, 5.20676786161221261625928265686, 5.72277992343933106741421004661, 6.70259218915186336018376994202, 7.57407683737923205314212000666, 8.786544768574228408243402550763, 9.410073394695902169016416531124

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