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Properties

Label 2-1911-1911.107-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.00106 - 0.999i$
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  + (0.365 + 0.930i)3-s + (−0.900 − 0.433i)4-s + (−0.733 − 0.680i)7-s + (−0.733 + 0.680i)9-s + (0.0747 − 0.997i)12-s + (0.826 + 0.563i)13-s + (0.623 + 0.781i)16-s + (0.733 + 1.26i)19-s + (0.365 − 0.930i)21-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.365 + 0.930i)28-s + (0.900 + 1.56i)31-s + (0.955 − 0.294i)36-s + (−1.48 + 0.716i)37-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)3-s + (−0.900 − 0.433i)4-s + (−0.733 − 0.680i)7-s + (−0.733 + 0.680i)9-s + (0.0747 − 0.997i)12-s + (0.826 + 0.563i)13-s + (0.623 + 0.781i)16-s + (0.733 + 1.26i)19-s + (0.365 − 0.930i)21-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.365 + 0.930i)28-s + (0.900 + 1.56i)31-s + (0.955 − 0.294i)36-s + (−1.48 + 0.716i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.00106 - 0.999i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.00106 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8550116287\)
\(L(\frac12)\) \(\approx\) \(0.8550116287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.365 - 0.930i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (-0.826 - 0.563i)T \)
good2 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.733 - 0.680i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.365 - 0.930i)T^{2} \)
31 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.48 - 0.716i)T + (0.623 - 0.781i)T^{2} \)
41 \( 1 + (-0.955 - 0.294i)T^{2} \)
43 \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.365 + 0.930i)T^{2} \)
73 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.619648312496885225354830852477, −8.942035695190716346984925614112, −8.325273861137399095319028677834, −7.34715002482446144633266345959, −6.20562126571454878762062512884, −5.49649471439295376138273930417, −4.56506736987318326450477322820, −3.77916826372569577275043443776, −3.25773150177681111197066438172, −1.44782910949169248013853990716, 0.66399908867029949101109191973, 2.39681476794583692702316043312, 3.18932605957308212060475661129, 4.05721210299585194337817861115, 5.35058531910701649705366442586, 6.02320315526124803698950879879, 6.89801850083319221944793439063, 7.83913882548472408119289508822, 8.344480140474226031523023673692, 9.240089777092894467840574798007

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