Properties

Label 2-260-260.127-c1-0-16
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: $\chi_{260} (127, \cdot )$
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

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

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