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Properties

Label 2-1911-1911.605-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.611 - 0.791i$
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  + (−0.680 − 0.733i)3-s + (−0.781 + 0.623i)4-s + (0.997 − 0.0747i)7-s + (−0.0747 + 0.997i)9-s + (0.988 + 0.149i)12-s + (−0.930 + 0.365i)13-s + (0.222 − 0.974i)16-s + (0.392 + 1.46i)19-s + (−0.733 − 0.680i)21-s + (−0.997 − 0.0747i)25-s + (0.781 − 0.623i)27-s + (−0.733 + 0.680i)28-s + (−0.0579 − 0.216i)31-s + (−0.563 − 0.826i)36-s + (0.205 + 1.82i)37-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)3-s + (−0.781 + 0.623i)4-s + (0.997 − 0.0747i)7-s + (−0.0747 + 0.997i)9-s + (0.988 + 0.149i)12-s + (−0.930 + 0.365i)13-s + (0.222 − 0.974i)16-s + (0.392 + 1.46i)19-s + (−0.733 − 0.680i)21-s + (−0.997 − 0.0747i)25-s + (0.781 − 0.623i)27-s + (−0.733 + 0.680i)28-s + (−0.0579 − 0.216i)31-s + (−0.563 − 0.826i)36-s + (0.205 + 1.82i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (605, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.611 - 0.791i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7013947219\)
\(L(\frac12)\) \(\approx\) \(0.7013947219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.680 + 0.733i)T \)
7 \( 1 + (-0.997 + 0.0747i)T \)
13 \( 1 + (0.930 - 0.365i)T \)
good2 \( 1 + (0.781 - 0.623i)T^{2} \)
5 \( 1 + (0.997 + 0.0747i)T^{2} \)
11 \( 1 + (-0.149 - 0.988i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (-0.392 - 1.46i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.733 - 0.680i)T^{2} \)
31 \( 1 + (0.0579 + 0.216i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.205 - 1.82i)T + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (-0.563 + 0.826i)T^{2} \)
43 \( 1 + (-0.587 - 1.90i)T + (-0.826 + 0.563i)T^{2} \)
47 \( 1 + (0.149 + 0.988i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.433 + 0.900i)T^{2} \)
61 \( 1 + (-1.84 + 0.722i)T + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.275 - 1.02i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.680 + 0.733i)T^{2} \)
73 \( 1 + (-0.900 + 0.774i)T + (0.149 - 0.988i)T^{2} \)
79 \( 1 + (-0.781 - 1.35i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.781 + 0.623i)T^{2} \)
89 \( 1 + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (1.26 + 0.337i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.614207452018355273325334496105, −8.371426049902508414744609621081, −7.933631508967947993257920655817, −7.39767769035238750773572639186, −6.34148305628671039637815796719, −5.36559230716774664081450085666, −4.75597833241237644641403117070, −3.89611699212873943392140850013, −2.49821270772090810901244055186, −1.32664542743279764674669519733, 0.63798162286424823449295512912, 2.24118658358144950886672764453, 3.78999527322318959678264019187, 4.54362540666494308415572708598, 5.30416740510075084714047122664, 5.61536165398819949275143285227, 6.87338233006101869795685117763, 7.74069261208821965648784451603, 8.839994677004148277276369463584, 9.279851762773383221302309462253

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