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Properties

Label 2-1911-273.233-c0-0-8
Degree $2$
Conductor $1911$
Sign $0.0725 - 0.997i$
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  + (−0.382 + 0.662i)2-s + (0.5 + 0.866i)3-s + (0.207 + 0.358i)4-s + (0.923 − 1.60i)5-s − 0.765·6-s − 1.08·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (0.382 + 0.662i)11-s + (−0.207 + 0.358i)12-s + 13-s + 1.84·15-s + (0.207 − 0.358i)16-s + (−0.382 − 0.662i)18-s + 0.765·20-s + ⋯
L(s)  = 1  + (−0.382 + 0.662i)2-s + (0.5 + 0.866i)3-s + (0.207 + 0.358i)4-s + (0.923 − 1.60i)5-s − 0.765·6-s − 1.08·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (0.382 + 0.662i)11-s + (−0.207 + 0.358i)12-s + 13-s + 1.84·15-s + (0.207 − 0.358i)16-s + (−0.382 − 0.662i)18-s + 0.765·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.0725 - 0.997i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.0725 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.392475648\)
\(L(\frac12)\) \(\approx\) \(1.392475648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
13 \( 1 - T \)
good2 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 1.84T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.84T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 0.765T + T^{2} \)
89 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.373014218733016135557259831216, −8.807270766671892655639402332706, −8.281591398530373979719945269937, −7.50175956019732741230562217438, −6.23147496980312313716906516493, −5.69742556797851821103595952392, −4.68926294939700567903562823121, −4.01182974707232069002535784578, −2.76447156525140656066759396207, −1.56197134847769259307107425229, 1.29221150763724240054510128039, 2.19940516658175250637691213852, 2.96491835321618998285552329909, 3.62789295559551307619892534488, 5.74974619976074426045120556436, 6.13465784303467825024004978656, 6.73791209646084992816265310331, 7.54438380924289958308154567672, 8.682534440788839760650830587829, 9.242348231451656799447215567759

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