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Properties

Label 2-1911-1911.155-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.718 + 0.695i$
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 + (0.900 + 0.433i)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 + (0.900 + 0.433i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.718 + 0.695i$
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: \((2,\ 1911,\ (\ :0),\ 0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.776564078\)
\(L(\frac12)\) \(\approx\) \(1.776564078\)
\(L(1)\) 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 \( 1 + (-0.900 - 0.433i)T \)
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

−9.249978974186229296489548301581, −8.523305555326470349092224662548, −7.88558131089529886729746751958, −6.94462137049488693397770526869, −6.05126293125960774214609657160, −5.51733308008598699959458055998, −4.29829218620317679489311199556, −3.33491112302346364596449495966, −1.98204630619652994907345182472, −1.57880841193952383045180349575, 1.67625767291194463698634079336, 2.88686341112339342896614129455, 3.57970090789448985897369888224, 4.39114245307915029501003394315, 5.16248065510141547902628701843, 6.74863419113681012677248369629, 7.20226066353901857683397735999, 8.191439363886392289506302363580, 8.491887583732759687228973183606, 9.255037275817857160204173756070

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