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Properties

Label 2-260-260.243-c1-0-22
Degree $2$
Conductor $260$
Sign $-0.172 + 0.984i$
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  + (−0.366 − 1.36i)2-s + (−1.73 + i)4-s + (1.86 − 1.23i)5-s + (2 + 1.99i)8-s + (2.59 − 1.5i)9-s + (−2.36 − 2.09i)10-s + (−0.232 − 3.59i)13-s + (1.99 − 3.46i)16-s + (−3.86 − 1.03i)17-s + (−3 − 3i)18-s + (−2 + 4i)20-s + (1.96 − 4.59i)25-s + (−4.83 + 1.63i)26-s + (5.76 + 3.33i)29-s + (−5.46 − 1.46i)32-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)4-s + (0.834 − 0.550i)5-s + (0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.748 − 0.663i)10-s + (−0.0643 − 0.997i)13-s + (0.499 − 0.866i)16-s + (−0.937 − 0.251i)17-s + (−0.707 − 0.707i)18-s + (−0.447 + 0.894i)20-s + (0.392 − 0.919i)25-s + (−0.947 + 0.320i)26-s + (1.07 + 0.618i)29-s + (−0.965 − 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.172 + 0.984i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763252 - 0.908961i\)
\(L(\frac12)\) \(\approx\) \(0.763252 - 0.908961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 + (-1.86 + 1.23i)T \)
13 \( 1 + (0.232 + 3.59i)T \)
good3 \( 1 + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.86 + 1.03i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-5.76 - 3.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (0.303 + 1.13i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.33 - 10.9i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (0.169 + 0.169i)T + 53iT^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.33 - 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.83 - 9.83i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (13.8 + 8i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (17.7 + 4.75i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.79279059924331991491966611783, −10.56483262238654445403616583066, −9.934656016936178970784909625278, −9.081425653133928842203279086269, −8.182670313163934404222519672721, −6.76196100470773627946624793768, −5.29982756770589490527677548566, −4.26406273546306389509324680586, −2.72329521910870590961641034107, −1.18967089992317893592939597282, 1.95100325114145574096345593038, 4.14966441261359599347151776550, 5.26798049548612350813225944030, 6.55723470307230794848023005047, 7.02685138958895284170952785180, 8.338545543240862976115667870552, 9.366380629836765256264651034796, 10.13795117572378348801571265323, 10.96895076562015835846652500132, 12.48762505097524592043518063489

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