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Properties

Label 2-1911-1911.1283-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.211 + 0.977i$
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  + (0.623 − 0.781i)3-s + (0.826 − 0.563i)4-s + (0.955 − 0.294i)7-s + (−0.222 − 0.974i)9-s + (0.0747 − 0.997i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)16-s − 1.46·19-s + (0.365 − 0.930i)21-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.623 − 0.781i)28-s + (0.900 + 1.56i)31-s + (−0.733 − 0.680i)36-s + (1.36 + 0.930i)37-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)3-s + (0.826 − 0.563i)4-s + (0.955 − 0.294i)7-s + (−0.222 − 0.974i)9-s + (0.0747 − 0.997i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)16-s − 1.46·19-s + (0.365 − 0.930i)21-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (0.623 − 0.781i)28-s + (0.900 + 1.56i)31-s + (−0.733 − 0.680i)36-s + (1.36 + 0.930i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.748700125\)
\(L(\frac12)\) \(\approx\) \(1.748700125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
good2 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (0.733 - 0.680i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + 1.46T + T^{2} \)
23 \( 1 + (0.988 - 0.149i)T^{2} \)
29 \( 1 + (0.988 + 0.149i)T^{2} \)
31 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.733 - 0.680i)T^{2} \)
43 \( 1 + (0.535 - 1.36i)T + (-0.733 - 0.680i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 - 1.24T + T^{2} \)
71 \( 1 + (0.988 - 0.149i)T^{2} \)
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 \( 1 + (-0.0747 + 0.997i)T^{2} \)
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

−9.197106461954208704146104665410, −8.147301906907679531262347505672, −7.76596284151270030143350840832, −6.81739090604159315399883387883, −6.36917644171108390669193076581, −5.23625242586575818780151941580, −4.34202979993813935706893671663, −2.97677039184548393952651338140, −2.12662620049210314573086269801, −1.30737627803931388769761302084, 2.26831189417063703491482252184, 2.38919720023663072200794696650, 3.86860441135761991976897973975, 4.44860080983232773093595827778, 5.47790333052658356262176063977, 6.41923461048516953283700687707, 7.54684285054464169911041034248, 8.025514474189591903376064106098, 8.609871659723388130001443421967, 9.579171621782339945802267484965

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