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Properties

Label 2-2132-2132.1995-c0-0-0
Degree $2$
Conductor $2132$
Sign $0.328 + 0.944i$
Analytic cond. $1.06400$
Root an. cond. $1.03150$
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)2-s + (−0.499 − 0.866i)4-s − 0.517i·5-s + 0.999·8-s + (−0.965 − 0.258i)9-s + (0.448 + 0.258i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.57 + 0.207i)17-s + (0.707 − 0.707i)18-s + (−0.448 + 0.258i)20-s + 0.732·25-s + (0.499 + 0.866i)26-s + (0.258 − 1.96i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.517i·5-s + 0.999·8-s + (−0.965 − 0.258i)9-s + (0.448 + 0.258i)10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.57 + 0.207i)17-s + (0.707 − 0.707i)18-s + (−0.448 + 0.258i)20-s + 0.732·25-s + (0.499 + 0.866i)26-s + (0.258 − 1.96i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2132\)    =    \(2^{2} \cdot 13 \cdot 41\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(1.06400\)
Root analytic conductor: \(1.03150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2132} (1995, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2132,\ (\ :0),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5307648298\)
\(L(\frac12)\) \(\approx\) \(0.5307648298\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
good3 \( 1 + (0.965 + 0.258i)T^{2} \)
5 \( 1 + 0.517iT - T^{2} \)
7 \( 1 + (0.965 - 0.258i)T^{2} \)
11 \( 1 + (0.258 - 0.965i)T^{2} \)
17 \( 1 + (1.57 - 0.207i)T + (0.965 - 0.258i)T^{2} \)
19 \( 1 + (-0.258 - 0.965i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.258 + 1.96i)T + (-0.965 - 0.258i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T^{2} \)
53 \( 1 + (0.758 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.965 + 0.258i)T^{2} \)
71 \( 1 + (-0.965 + 0.258i)T^{2} \)
73 \( 1 + 1.93iT - T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
97 \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \)
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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

−8.833140175758636249687655444447, −8.429199679276550951031736021181, −7.80025593584682974119654156816, −6.58661409192467349630160497115, −6.20475081167958039363240999600, −5.23189003002292715255967275356, −4.58334136661986966415341431697, −3.38245490440612209380406822679, −1.99146437139464298969180646845, −0.43347235725537391182711991725, 1.61696020669112714916462698573, 2.64062998642737766375979709307, 3.37767104627508793148038153140, 4.42900108398108660789328675041, 5.26470438353438393594483758751, 6.60676953173773567331067011147, 7.03853755637929538482558190986, 8.216183995833714877534414215970, 8.857496821375262579590967804241, 9.216566291845774842250584040894

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