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Properties

Label 2-260-13.4-c1-0-1
Degree $2$
Conductor $260$
Sign $0.916 - 0.400i$
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  + (0.0473 − 0.0820i)3-s + i·5-s + (0.716 − 0.413i)7-s + (1.49 + 2.59i)9-s + (1.5 + 0.866i)11-s + (3.32 − 1.40i)13-s + (0.0820 + 0.0473i)15-s + (0.716 + 1.24i)17-s + (−0.926 + 0.534i)19-s − 0.0783i·21-s + (1.54 − 2.67i)23-s − 25-s + 0.567·27-s + (−3.72 + 6.45i)29-s − 5.84i·31-s + ⋯
L(s)  = 1  + (0.0273 − 0.0473i)3-s + 0.447i·5-s + (0.270 − 0.156i)7-s + (0.498 + 0.863i)9-s + (0.452 + 0.261i)11-s + (0.921 − 0.388i)13-s + (0.0211 + 0.0122i)15-s + (0.173 + 0.300i)17-s + (−0.212 + 0.122i)19-s − 0.0171i·21-s + (0.321 − 0.557i)23-s − 0.200·25-s + 0.109·27-s + (−0.692 + 1.19i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34660 + 0.281263i\)
\(L(\frac12)\) \(\approx\) \(1.34660 + 0.281263i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.32 + 1.40i)T \)
good3 \( 1 + (-0.0473 + 0.0820i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.716 + 0.413i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.716 - 1.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.926 - 0.534i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.54 + 2.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.72 - 6.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.84iT - 31T^{2} \)
37 \( 1 + (-0.851 - 0.491i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.69 + 2.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.77 + 8.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 0.334T + 53T^{2} \)
59 \( 1 + (9.98 - 5.76i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.35 + 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.9 + 6.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.46 + 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 + 0.252T + 79T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (-3.98 - 2.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.25 + 4.76i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.03175988299041390297454249237, −10.85327841083357220980102835346, −10.45806954086209175152344134340, −9.146592567455175889442491432703, −8.068766365330817519317633891550, −7.17818191219213020635138124651, −6.06386400431679203068541683356, −4.76084435262839747445338023786, −3.50528892861905740085357026812, −1.79395502158262866374458092708, 1.38241620604394333990585995176, 3.43960957350700788253176807088, 4.55419633479545802237819638477, 5.91106755076808276128671448685, 6.87322533062007821432925684744, 8.181359639823472124613913810683, 9.084357610153755044413660160724, 9.832021088356583630332103060534, 11.18981923110417809546115770882, 11.84049490929472970409911396743

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