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Properties

Label 2-260-260.103-c1-0-2
Degree $2$
Conductor $260$
Sign $-0.933 + 0.359i$
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  + (0.178 + 1.40i)2-s + (−1.93 + 0.5i)4-s + (−1.58 − 1.58i)5-s + (−2.44 + 2.44i)7-s + (−1.04 − 2.62i)8-s + 3i·9-s + (1.93 − 2.50i)10-s − 2.44·11-s + (−3.58 − 0.418i)13-s + (−3.87 − 3i)14-s + (3.50 − 1.93i)16-s + (−1 + i)17-s + (−4.20 + 0.534i)18-s + 2.44i·19-s + (3.85 + 2.27i)20-s + ⋯
L(s)  = 1  + (0.126 + 0.992i)2-s + (−0.968 + 0.250i)4-s + (−0.707 − 0.707i)5-s + (−0.925 + 0.925i)7-s + (−0.370 − 0.929i)8-s + i·9-s + (0.612 − 0.790i)10-s − 0.738·11-s + (−0.993 − 0.116i)13-s + (−1.03 − 0.801i)14-s + (0.875 − 0.484i)16-s + (−0.242 + 0.242i)17-s + (−0.992 + 0.126i)18-s + 0.561i·19-s + (0.861 + 0.507i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.933 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0771548 - 0.414496i\)
\(L(\frac12)\) \(\approx\) \(0.0771548 - 0.414496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.178 - 1.40i)T \)
5 \( 1 + (1.58 + 1.58i)T \)
13 \( 1 + (3.58 + 0.418i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (2.44 - 2.44i)T - 7iT^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (-3.87 + 3.87i)T - 23iT^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 9.79T + 31T^{2} \)
37 \( 1 + (6.32 + 6.32i)T + 37iT^{2} \)
41 \( 1 - 6.32iT - 41T^{2} \)
43 \( 1 + (7.74 - 7.74i)T - 43iT^{2} \)
47 \( 1 + (2.44 - 2.44i)T - 47iT^{2} \)
53 \( 1 + (-2 - 2i)T + 53iT^{2} \)
59 \( 1 + 7.34iT - 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (2.44 - 2.44i)T - 67iT^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + (3.16 - 3.16i)T - 73iT^{2} \)
79 \( 1 + 7.74T + 79T^{2} \)
83 \( 1 + (-4.89 - 4.89i)T + 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \)
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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

−12.75185799796562336998481903858, −11.94536983983910590697357918415, −10.39912964799828430838346740262, −9.387184195612088759905427495279, −8.400043635250948823277015203649, −7.76665433369871363488031484345, −6.56003107753701338214378746046, −5.31274471540725928290364530209, −4.63293773334837399421387742549, −2.93865623041988867122046125664, 0.30605336804112161333234131524, 2.84655685631853651544109379918, 3.64443011260866052772093449721, 4.85791199641906049639367483052, 6.56838122233126039265198161959, 7.42481928343096874347202824776, 8.815691365461588921010508338366, 9.981529073091620941241576797378, 10.37359757592816054092946970769, 11.60202449713629255990435808849

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