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Properties

Label 2-260-260.187-c0-0-0
Degree $2$
Conductor $260$
Sign $0.749 + 0.661i$
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  − 2-s + 4-s − i·5-s − 8-s − i·9-s + i·10-s + i·13-s + 16-s + (1 − i)17-s + i·18-s − i·20-s − 25-s − i·26-s + 2i·29-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s − i·5-s − 8-s − i·9-s + i·10-s + i·13-s + 16-s + (1 − i)17-s + i·18-s − i·20-s − 25-s − i·26-s + 2i·29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(0.129756\)
Root analytic conductor: \(0.360217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :0),\ 0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5059126929\)
\(L(\frac12)\) \(\approx\) \(0.5059126929\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + iT \)
13 \( 1 - iT \)
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 \( 1 - 2T + T^{2} \)
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

−12.03868078674636152471222880535, −11.20980221048105044725047029261, −9.815536100597621499556067821481, −9.274106424248876804879528439435, −8.478296824641379647754730030440, −7.32346580397877620284854966713, −6.32641308821268774433648266607, −5.00857670055161515029135103719, −3.33858863379832676236069890591, −1.37736054286372667276825657430, 2.16114574364275784524204190894, 3.44649809219711510079715086898, 5.53657516302876886290621075646, 6.53443278402019860554475465853, 7.81045689213456306064817317062, 8.109643467058337085922005046852, 9.749092543078072203135029193729, 10.37317329210454955630126998659, 11.04239992511272870701939745140, 12.02582929086622292141981981243

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