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Properties

Label 2-1911-1911.1115-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.343 + 0.939i$
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  + (−0.955 − 0.294i)3-s + (−0.623 − 0.781i)4-s + (−0.826 + 0.563i)7-s + (0.826 + 0.563i)9-s + (0.365 + 0.930i)12-s + (0.988 + 0.149i)13-s + (−0.222 + 0.974i)16-s + (0.975 − 0.563i)19-s + (0.955 − 0.294i)21-s + (−0.826 − 0.563i)25-s + (−0.623 − 0.781i)27-s + (0.955 + 0.294i)28-s + (1.35 − 0.781i)31-s + (−0.0747 − 0.997i)36-s + (0.233 + 0.185i)37-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)3-s + (−0.623 − 0.781i)4-s + (−0.826 + 0.563i)7-s + (0.826 + 0.563i)9-s + (0.365 + 0.930i)12-s + (0.988 + 0.149i)13-s + (−0.222 + 0.974i)16-s + (0.975 − 0.563i)19-s + (0.955 − 0.294i)21-s + (−0.826 − 0.563i)25-s + (−0.623 − 0.781i)27-s + (0.955 + 0.294i)28-s + (1.35 − 0.781i)31-s + (−0.0747 − 0.997i)36-s + (0.233 + 0.185i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6278046878\)
\(L(\frac12)\) \(\approx\) \(0.6278046878\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.955 + 0.294i)T \)
7 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
good2 \( 1 + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (0.826 + 0.563i)T^{2} \)
11 \( 1 + (0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (-0.975 + 0.563i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.955 - 0.294i)T^{2} \)
31 \( 1 + (-1.35 + 0.781i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.233 - 0.185i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.0747 - 0.997i)T^{2} \)
43 \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \)
47 \( 1 + (0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.900 - 0.433i)T^{2} \)
61 \( 1 + (-0.722 - 0.108i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (1.68 + 0.974i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.955 - 0.294i)T^{2} \)
73 \( 1 + (-1.04 + 1.53i)T + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (-0.975 - 0.563i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.382972421394630126978474247952, −8.606094363463844531545321408537, −7.61209283942725368611192653688, −6.41071406362586548955670846746, −6.19288216863384484939574330636, −5.32998284311281547002704802249, −4.56468348686401720585184141379, −3.49170587854751218094142765509, −2.00356624146118324470404529965, −0.68047020158423249863973022884, 1.08097618231577545309076783331, 3.17028243499364303267702512278, 3.76171789087851496312391327624, 4.58722838826019355049600526185, 5.54311296290914915051933437792, 6.34950771514686223745088134799, 7.14307771623791306098857621938, 7.946405903772052691497304485633, 8.828562469803658111120424101283, 9.809614464042706089192865061992

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