Properties

Label 2-260-13.12-c1-0-1
Degree $2$
Conductor $260$
Sign $0.183 - 0.983i$
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  − 0.339·3-s + i·5-s + 3.88i·7-s − 2.88·9-s + 1.54i·11-s + (3.54 + 0.660i)13-s − 0.339i·15-s + 2.86i·19-s − 1.32i·21-s + 5.42·23-s − 25-s + 2·27-s − 5.20·29-s + 6.22i·31-s − 0.524i·33-s + ⋯
L(s)  = 1  − 0.196·3-s + 0.447i·5-s + 1.46i·7-s − 0.961·9-s + 0.465i·11-s + (0.983 + 0.183i)13-s − 0.0877i·15-s + 0.657i·19-s − 0.288i·21-s + 1.13·23-s − 0.200·25-s + 0.384·27-s − 0.966·29-s + 1.11i·31-s − 0.0913i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.183 - 0.983i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.816246 + 0.678268i\)
\(L(\frac12)\) \(\approx\) \(0.816246 + 0.678268i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-3.54 - 0.660i)T \)
good3 \( 1 + 0.339T + 3T^{2} \)
7 \( 1 - 3.88iT - 7T^{2} \)
11 \( 1 - 1.54iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.86iT - 19T^{2} \)
23 \( 1 - 5.42T + 23T^{2} \)
29 \( 1 + 5.20T + 29T^{2} \)
31 \( 1 - 6.22iT - 31T^{2} \)
37 \( 1 + 8.56iT - 37T^{2} \)
41 \( 1 + 9.08iT - 41T^{2} \)
43 \( 1 - 0.980T + 43T^{2} \)
47 \( 1 + 6.52iT - 47T^{2} \)
53 \( 1 - 6.44T + 53T^{2} \)
59 \( 1 - 4.45iT - 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 6.97iT - 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + 3.43iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 8.56iT - 83T^{2} \)
89 \( 1 + 17.1iT - 89T^{2} \)
97 \( 1 - 13.7iT - 97T^{2} \)
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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

−12.06479032731554021374154377572, −11.34564934986328715922268367728, −10.46303123223492261551361521557, −9.046120503917930060473072680772, −8.626837606767059098802184003958, −7.18723453642744003785268569533, −5.97334070312009896843874015217, −5.33706374066850246774696629849, −3.52194865543535629673090087107, −2.22453243916644613698578873050, 0.894464998644653211329237356339, 3.20869582271191960041843794520, 4.42694601665224417265258606945, 5.67626480644896880904916524832, 6.76723577313674734978808031095, 7.937729835859477063608209308651, 8.806296291302671338372418577886, 9.936873445749874374708830575235, 11.19406063249565506796310157420, 11.30438393746008789850740416108

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