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Properties

Label 2-260-260.47-c2-0-45
Degree $2$
Conductor $260$
Sign $0.486 + 0.873i$
Analytic cond. $7.08448$
Root an. cond. $2.66166$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 4·4-s + (−4 + 3i)5-s − 8i·8-s − 9i·9-s + (−6 − 8i)10-s + (−12 − 5i)13-s + 16·16-s + (7 − 7i)17-s + 18·18-s + (16 − 12i)20-s + (7 − 24i)25-s + (10 − 24i)26-s − 42i·29-s + 32i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.800 + 0.600i)5-s − i·8-s − i·9-s + (−0.600 − 0.800i)10-s + (−0.923 − 0.384i)13-s + 16-s + (0.411 − 0.411i)17-s + 18-s + (0.800 − 0.600i)20-s + (0.280 − 0.959i)25-s + (0.384 − 0.923i)26-s − 1.44i·29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(7.08448\)
Root analytic conductor: \(2.66166\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.418721 - 0.246043i\)
\(L(\frac12)\) \(\approx\) \(0.418721 - 0.246043i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (4 - 3i)T \)
13 \( 1 + (12 + 5i)T \)
good3 \( 1 + 9iT^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 + 121iT^{2} \)
17 \( 1 + (-7 + 7i)T - 289iT^{2} \)
19 \( 1 + 361iT^{2} \)
23 \( 1 - 529iT^{2} \)
29 \( 1 + 42iT - 841T^{2} \)
31 \( 1 - 961iT^{2} \)
37 \( 1 + 70T + 1.36e3T^{2} \)
41 \( 1 + (31 + 31i)T + 1.68e3iT^{2} \)
43 \( 1 - 1.84e3iT^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + (17 - 17i)T - 2.80e3iT^{2} \)
59 \( 1 - 3.48e3iT^{2} \)
61 \( 1 + 120T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3iT^{2} \)
73 \( 1 - 96iT - 5.32e3T^{2} \)
79 \( 1 + 6.24e3T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (119 + 119i)T + 7.92e3iT^{2} \)
97 \( 1 - 130iT - 9.40e3T^{2} \)
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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

−11.90245657011630371666034301563, −10.43660431704034418488378087627, −9.564926566264003800601148542970, −8.482005977539628466310254078492, −7.47537904972169583062566800953, −6.80783265711327037859865440150, −5.63857006014439455264218283381, −4.31982551619498085341767004819, −3.20613547935652348268946761115, −0.25486356112530253149632986021, 1.68318336795903669521374100774, 3.27245606723445605414005142344, 4.56977968768270041772575005928, 5.28743547575822780141689867845, 7.29323845325303099955400953976, 8.252233569934598457847089473189, 9.083315930621628348736480468206, 10.24001818322834031860981731517, 11.00110610362271524497237542117, 12.03814936354519385069162048355

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