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Properties

Label 2-260-20.7-c1-0-4
Degree $2$
Conductor $260$
Sign $-0.441 - 0.897i$
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  + (−1.39 + 0.203i)2-s + (0.0775 + 0.0775i)3-s + (1.91 − 0.569i)4-s + (0.871 + 2.05i)5-s + (−0.124 − 0.0927i)6-s + (−2.89 + 2.89i)7-s + (−2.56 + 1.18i)8-s − 2.98i·9-s + (−1.63 − 2.70i)10-s + 3.20i·11-s + (0.192 + 0.104i)12-s + (−0.707 + 0.707i)13-s + (3.46 − 4.64i)14-s + (−0.0921 + 0.227i)15-s + (3.35 − 2.18i)16-s + (−0.217 − 0.217i)17-s + ⋯
L(s)  = 1  + (−0.989 + 0.143i)2-s + (0.0447 + 0.0447i)3-s + (0.958 − 0.284i)4-s + (0.389 + 0.920i)5-s + (−0.0507 − 0.0378i)6-s + (−1.09 + 1.09i)7-s + (−0.907 + 0.419i)8-s − 0.995i·9-s + (−0.518 − 0.855i)10-s + 0.965i·11-s + (0.0556 + 0.0301i)12-s + (−0.196 + 0.196i)13-s + (0.925 − 1.24i)14-s + (−0.0237 + 0.0586i)15-s + (0.837 − 0.545i)16-s + (−0.0528 − 0.0528i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.441 - 0.897i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340909 + 0.547635i\)
\(L(\frac12)\) \(\approx\) \(0.340909 + 0.547635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.203i)T \)
5 \( 1 + (-0.871 - 2.05i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.0775 - 0.0775i)T + 3iT^{2} \)
7 \( 1 + (2.89 - 2.89i)T - 7iT^{2} \)
11 \( 1 - 3.20iT - 11T^{2} \)
17 \( 1 + (0.217 + 0.217i)T + 17iT^{2} \)
19 \( 1 + 5.01T + 19T^{2} \)
23 \( 1 + (-4.02 - 4.02i)T + 23iT^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 + 2.20iT - 31T^{2} \)
37 \( 1 + (-0.278 - 0.278i)T + 37iT^{2} \)
41 \( 1 - 5.39T + 41T^{2} \)
43 \( 1 + (1.36 + 1.36i)T + 43iT^{2} \)
47 \( 1 + (-4.21 + 4.21i)T - 47iT^{2} \)
53 \( 1 + (0.797 - 0.797i)T - 53iT^{2} \)
59 \( 1 - 9.66T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + (-3.68 + 3.68i)T - 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-0.326 + 0.326i)T - 73iT^{2} \)
79 \( 1 + 6.73T + 79T^{2} \)
83 \( 1 + (2.35 + 2.35i)T + 83iT^{2} \)
89 \( 1 - 16.4iT - 89T^{2} \)
97 \( 1 + (11.5 + 11.5i)T + 97iT^{2} \)
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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

−12.20437504184420768794434691043, −11.16258321082388983318648564441, −10.09073457305262261316071078672, −9.430223963569689647602548947124, −8.791525895734895829713446292241, −7.12401306362341625600384788889, −6.61771916026832978694736338953, −5.64229057822653370493736237745, −3.32782051763899110946571851350, −2.21839279904044213876838827020, 0.65854889590269779436018344529, 2.53667288691574167153759071589, 4.16288224702737806955097259474, 5.83716119624689130523574745174, 6.85172717739159314231955404874, 8.028202317266616105705397520433, 8.755666231569707963239804065108, 9.867976901645904832470093212240, 10.47954889723841919368303792300, 11.39143702754140517410137262622

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