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Properties

Label 2-1911-1911.155-c0-0-1
Degree $2$
Conductor $1911$
Sign $-0.0960 + 0.995i$
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  + (0.900 − 0.433i)3-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)7-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)12-s + (0.222 + 0.974i)13-s + (−0.900 − 0.433i)16-s − 1.56i·19-s + (−0.900 − 0.433i)21-s + (−0.623 + 0.781i)25-s + (0.222 − 0.974i)27-s + (−0.900 + 0.433i)28-s + 1.94i·31-s + (−0.623 − 0.781i)36-s + (1.90 − 0.433i)37-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)3-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)7-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)12-s + (0.222 + 0.974i)13-s + (−0.900 − 0.433i)16-s − 1.56i·19-s + (−0.900 − 0.433i)21-s + (−0.623 + 0.781i)25-s + (0.222 − 0.974i)27-s + (−0.900 + 0.433i)28-s + 1.94i·31-s + (−0.623 − 0.781i)36-s + (1.90 − 0.433i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.0960 + 0.995i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.0960 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.515631606\)
\(L(\frac12)\) \(\approx\) \(1.515631606\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
good2 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + 1.56iT - T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 - 1.94iT - T^{2} \)
37 \( 1 + (-1.90 + 0.433i)T + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + 0.867iT - T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-1.52 - 1.21i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.222 - 0.974i)T^{2} \)
97 \( 1 - 1.56iT - T^{2} \)
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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

−9.436296129382055593441199721058, −8.534047539835964864155380770645, −7.42006936744372663117794409491, −6.80048456743224614033149547709, −6.38196967274569625719250898235, −5.08605135466787474688935759262, −4.17091088540413052473144852884, −3.17762609995467602014556697767, −2.12403608556198848560967109353, −1.04410312776987380221359488402, 2.10835340372533395346812372284, 2.89353341551948551713759688849, 3.63509601488902122202826137377, 4.39564521279826016417074051495, 5.71003317437792505991641512551, 6.40925065741729075883576105414, 7.72525608775641331517616863545, 7.997839935950560394199926962929, 8.667989523415485058847091571528, 9.667487012797143965708203496837

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