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Properties

Label 2-1911-1911.1865-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.790 + 0.611i$
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  + (0.563 − 0.826i)3-s + (0.974 + 0.222i)4-s + (0.930 + 0.365i)7-s + (−0.365 − 0.930i)9-s + (0.733 − 0.680i)12-s + (0.294 − 0.955i)13-s + (0.900 + 0.433i)16-s + (−1.77 + 0.474i)19-s + (0.826 − 0.563i)21-s + (−0.930 + 0.365i)25-s + (−0.974 − 0.222i)27-s + (0.826 + 0.563i)28-s + (−1.02 + 0.275i)31-s + (−0.149 − 0.988i)36-s + (−0.497 + 0.791i)37-s + ⋯
L(s)  = 1  + (0.563 − 0.826i)3-s + (0.974 + 0.222i)4-s + (0.930 + 0.365i)7-s + (−0.365 − 0.930i)9-s + (0.733 − 0.680i)12-s + (0.294 − 0.955i)13-s + (0.900 + 0.433i)16-s + (−1.77 + 0.474i)19-s + (0.826 − 0.563i)21-s + (−0.930 + 0.365i)25-s + (−0.974 − 0.222i)27-s + (0.826 + 0.563i)28-s + (−1.02 + 0.275i)31-s + (−0.149 − 0.988i)36-s + (−0.497 + 0.791i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.790 + 0.611i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.790 + 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.821369024\)
\(L(\frac12)\) \(\approx\) \(1.821369024\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.563 + 0.826i)T \)
7 \( 1 + (-0.930 - 0.365i)T \)
13 \( 1 + (-0.294 + 0.955i)T \)
good2 \( 1 + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (0.930 - 0.365i)T^{2} \)
11 \( 1 + (-0.680 + 0.733i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + (1.77 - 0.474i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.826 - 0.563i)T^{2} \)
31 \( 1 + (1.02 - 0.275i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.497 - 0.791i)T + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (-0.149 + 0.988i)T^{2} \)
43 \( 1 + (-1.85 + 0.139i)T + (0.988 - 0.149i)T^{2} \)
47 \( 1 + (0.680 - 0.733i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.781 + 0.623i)T^{2} \)
61 \( 1 + (0.432 - 1.40i)T + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.638 + 0.170i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.563 + 0.826i)T^{2} \)
73 \( 1 + (-0.0685 - 0.0299i)T + (0.680 + 0.733i)T^{2} \)
79 \( 1 + (0.974 + 1.68i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.974 + 0.222i)T^{2} \)
89 \( 1 + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (-0.206 + 0.772i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.949782683163567556383204781070, −8.392396626967934159540533257010, −7.72188358038928089001311512136, −7.18816045321298065687149114424, −6.09816823771398572205685449556, −5.69445562930912662576302113920, −4.20142777722057992332094907956, −3.18028355039367541966065837255, −2.23493266219408629907806979408, −1.52624268565872018925878806647, 1.82947076807366493047818492033, 2.41784295377711993191563378332, 3.83017396694054309800784246215, 4.37526024879581129593465761343, 5.42472434916379878429951207102, 6.30492703795662844902999680184, 7.24845614337954338737029133804, 7.938258034770774899220332190405, 8.735365332197737737784715977352, 9.438241598699293994413930359308

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