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Properties

Label 2-260-65.57-c1-0-2
Degree $2$
Conductor $260$
Sign $0.983 - 0.180i$
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  + (0.575 − 0.575i)3-s + (2.16 + 0.575i)5-s + 2.48i·7-s + 2.33i·9-s + (−0.160 − 0.160i)11-s + (1.42 − 3.31i)13-s + (1.57 − 0.912i)15-s + (1.58 − 1.58i)17-s + (−2.91 − 2.91i)19-s + (1.43 + 1.43i)21-s + (−0.839 − 0.839i)23-s + (4.33 + 2.48i)25-s + (3.07 + 3.07i)27-s − 2.80i·29-s + (−3.31 + 3.31i)31-s + ⋯
L(s)  = 1  + (0.332 − 0.332i)3-s + (0.966 + 0.257i)5-s + 0.940i·7-s + 0.778i·9-s + (−0.0484 − 0.0484i)11-s + (0.394 − 0.918i)13-s + (0.406 − 0.235i)15-s + (0.384 − 0.384i)17-s + (−0.668 − 0.668i)19-s + (0.312 + 0.312i)21-s + (−0.175 − 0.175i)23-s + (0.867 + 0.497i)25-s + (0.591 + 0.591i)27-s − 0.521i·29-s + (−0.594 + 0.594i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.983 - 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58305 + 0.143702i\)
\(L(\frac12)\) \(\approx\) \(1.58305 + 0.143702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.16 - 0.575i)T \)
13 \( 1 + (-1.42 + 3.31i)T \)
good3 \( 1 + (-0.575 + 0.575i)T - 3iT^{2} \)
7 \( 1 - 2.48iT - 7T^{2} \)
11 \( 1 + (0.160 + 0.160i)T + 11iT^{2} \)
17 \( 1 + (-1.58 + 1.58i)T - 17iT^{2} \)
19 \( 1 + (2.91 + 2.91i)T + 19iT^{2} \)
23 \( 1 + (0.839 + 0.839i)T + 23iT^{2} \)
29 \( 1 + 2.80iT - 29T^{2} \)
31 \( 1 + (3.31 - 3.31i)T - 31iT^{2} \)
37 \( 1 + 5.33iT - 37T^{2} \)
41 \( 1 + (6.07 - 6.07i)T - 41iT^{2} \)
43 \( 1 + (7.48 + 7.48i)T + 43iT^{2} \)
47 \( 1 + 3.99iT - 47T^{2} \)
53 \( 1 + (0.0154 - 0.0154i)T - 53iT^{2} \)
59 \( 1 + (2.91 - 2.91i)T - 59iT^{2} \)
61 \( 1 + 7.60T + 61T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + (11.3 - 11.3i)T - 71iT^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 15.6iT - 79T^{2} \)
83 \( 1 - 0.681iT - 83T^{2} \)
89 \( 1 + (8.11 - 8.11i)T - 89iT^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−12.15627563786751209997884214886, −10.92049963912446831501431251033, −10.17094819793512313886529234013, −9.018092861186128906525045049666, −8.265829655509699247126451067980, −7.05273916635555774935137856604, −5.87123032487969293973946769115, −5.07036635060196351533026050046, −3.01048754699987422437820009950, −1.99177445028314580724706508339, 1.59446527235327441419065788759, 3.47270713777382595923307941713, 4.53114807776115897715715771494, 6.00860660122958612373343071390, 6.84164995836433058602363645606, 8.254913492903047199008883397522, 9.231982970886972722552577463655, 9.970227191972492474758681393155, 10.77772589661813631811283261901, 12.04022518887376369621182889682

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