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Properties

Label 2-260-260.207-c1-0-12
Degree $2$
Conductor $260$
Sign $0.565 - 0.824i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.178 + 1.40i)2-s + (−1.93 − 0.5i)4-s + (1.58 − 1.58i)5-s + (2.44 + 2.44i)7-s + (1.04 − 2.62i)8-s − 3i·9-s + (1.93 + 2.50i)10-s + 2.44·11-s + (−0.418 + 3.58i)13-s + (−3.87 + 3i)14-s + (3.50 + 1.93i)16-s + (−1 − i)17-s + (4.20 + 0.534i)18-s + 2.44i·19-s + (−3.85 + 2.27i)20-s + ⋯
L(s)  = 1  + (−0.126 + 0.992i)2-s + (−0.968 − 0.250i)4-s + (0.707 − 0.707i)5-s + (0.925 + 0.925i)7-s + (0.370 − 0.929i)8-s − i·9-s + (0.612 + 0.790i)10-s + 0.738·11-s + (−0.116 + 0.993i)13-s + (−1.03 + 0.801i)14-s + (0.875 + 0.484i)16-s + (−0.242 − 0.242i)17-s + (0.992 + 0.126i)18-s + 0.561i·19-s + (−0.861 + 0.507i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.565 - 0.824i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.565 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17357 + 0.618349i\)
\(L(\frac12)\) \(\approx\) \(1.17357 + 0.618349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.178 - 1.40i)T \)
5 \( 1 + (-1.58 + 1.58i)T \)
13 \( 1 + (0.418 - 3.58i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (-3.87 - 3.87i)T + 23iT^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 9.79T + 31T^{2} \)
37 \( 1 + (-6.32 + 6.32i)T - 37iT^{2} \)
41 \( 1 - 6.32iT - 41T^{2} \)
43 \( 1 + (7.74 + 7.74i)T + 43iT^{2} \)
47 \( 1 + (-2.44 - 2.44i)T + 47iT^{2} \)
53 \( 1 + (-2 + 2i)T - 53iT^{2} \)
59 \( 1 + 7.34iT - 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-2.44 - 2.44i)T + 67iT^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + (-3.16 - 3.16i)T + 73iT^{2} \)
79 \( 1 + 7.74T + 79T^{2} \)
83 \( 1 + (4.89 - 4.89i)T - 83iT^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (3.16 - 3.16i)T - 97iT^{2} \)
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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

−12.24122099317061591450490047475, −11.35557026957247131310864018682, −9.591643391292293954800961368064, −9.177727662264025988721555931437, −8.451839188183648529874603915748, −7.09086269540284467417999595148, −6.03064305937472311179600113266, −5.22677961236133929375591998199, −4.05977718088841325333421298362, −1.59227403407412329797455187407, 1.54820177126435894202870823023, 2.91026128516531628028517714482, 4.38676027525123866439003553849, 5.41244736634111049401996388134, 7.08016801492524016624396240964, 8.051311521455995268834692773103, 9.166600130951731187719085524977, 10.38063550355680938713603827306, 10.76294749324272330517961577360, 11.46971687135094154311329229558

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