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Properties

Label 2-260-260.43-c1-0-10
Degree $2$
Conductor $260$
Sign $0.872 - 0.488i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (1.86 + 1.23i)5-s + (−1.99 + 2i)8-s + (2.59 − 1.5i)9-s + (−3 − 0.999i)10-s + (−3.59 + 0.232i)13-s + (1.99 − 3.46i)16-s + (6.96 + 1.86i)17-s + (−3 + 3i)18-s + (4.46 + 0.267i)20-s + (1.96 + 4.59i)25-s + (4.83 − 1.63i)26-s + (5.76 + 3.33i)29-s + (−1.46 + 5.46i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)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 + (−0.948 − 0.316i)10-s + (−0.997 + 0.0643i)13-s + (0.499 − 0.866i)16-s + (1.68 + 0.452i)17-s + (−0.707 + 0.707i)18-s + (0.998 + 0.0599i)20-s + (0.392 + 0.919i)25-s + (0.947 − 0.320i)26-s + (1.07 + 0.618i)29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.968320 + 0.252881i\)
\(L(\frac12)\) \(\approx\) \(0.968320 + 0.252881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 + (-1.86 - 1.23i)T \)
13 \( 1 + (3.59 - 0.232i)T \)
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 \( 1 + (-6.96 - 1.86i)T + (14.7 + 8.5i)T^{2} \)
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 \( 1 + (1.13 - 0.303i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.66 - 0.964i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (10.2 + 10.2i)T + 53iT^{2} \)
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 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.75 + 17.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.99089674073266853191756718723, −10.72915262068797434024087407992, −9.902142723296848927879469962320, −9.549871022256300093844127359665, −8.131400654565639397295644617651, −7.13353433799032811832697317839, −6.35180832497297740861400420421, −5.17643850111345254423070282768, −3.10735196321887568127731194372, −1.55798000021671217553507804455, 1.35343866750416102201107010287, 2.77270746314136231897448015067, 4.68089880556546600524935083596, 5.96617526425806339880635493458, 7.27866651381943936682065003800, 8.031606914787126180474967362253, 9.305378670186037394591217865844, 9.930593692676980040845706624641, 10.56600482245953550515964832593, 12.03416659979469693701115666004

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