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Properties

Label 2-260-260.83-c0-0-0
Degree $2$
Conductor $260$
Sign $0.256 - 0.966i$
Analytic cond. $0.129756$
Root an. cond. $0.360217$
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  + i·2-s − 4-s + 5-s − i·8-s + i·9-s + i·10-s − 13-s + 16-s + (−1 − i)17-s − 18-s − 20-s + 25-s − i·26-s − 2i·29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + 5-s − i·8-s + i·9-s + i·10-s − 13-s + 16-s + (−1 − i)17-s − 18-s − 20-s + 25-s − i·26-s − 2i·29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.256 - 0.966i$
Analytic conductor: \(0.129756\)
Root analytic conductor: \(0.360217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :0),\ 0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7529341310\)
\(L(\frac12)\) \(\approx\) \(0.7529341310\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 - iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
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

−12.82251058420569389269467634263, −11.45088335021371212431244632391, −10.11698450147350455664565324277, −9.558298903833028026275004216819, −8.446478796280666770845829155343, −7.41235402654655989524850721661, −6.46760654973349580082843120313, −5.32889621241390485713710346214, −4.56557210181478800039033333937, −2.43061537449341308707134139626, 1.81103030117855911204439404581, 3.19491018413712777771327149791, 4.60645839685577687682105807730, 5.77247295088478574941748985570, 6.97217871667529371634475740297, 8.674766221919918008550416064948, 9.269196707780356329923341568476, 10.21572661603365946786259719149, 10.92493001509926890157433226094, 12.23150478996913530495027146426

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