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Properties

Label 2-912-57.11-c0-0-0
Degree $2$
Conductor $912$
Sign $0.671 + 0.740i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (0.5 − 0.866i)3-s + 7-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)13-s − 19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)25-s − 0.999·27-s + 31-s − 37-s + 0.999·39-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)57-s + (0.5 + 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + 7-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)13-s − 19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)25-s − 0.999·27-s + 31-s − 37-s + 0.999·39-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)57-s + (0.5 + 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.671 + 0.740i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251908771\)
\(L(\frac12)\) \(\approx\) \(1.251908771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.18564362428135691475151304906, −9.071560745922779907350886428329, −8.395069155310954657775236370113, −7.83193173136075888241536855645, −6.75368989737793507852430507160, −6.13454410774604462355546476151, −4.81220028357752716294396926121, −3.83572741687139553627097730059, −2.43332388812986961015270571047, −1.49051568269942084599327398276, 1.87263121972076593513080425191, 3.15760897999164346984201921690, 4.15222513866735705651646429404, 5.03847529746151832884127132371, 5.83919761185662911970079025778, 7.19204268638908730022330304421, 8.314488287632337471742068424673, 8.468149385590209130116238328543, 9.632322595847121007642387963406, 10.43556390435928083917637937386

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