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Properties

Label 2-260-260.147-c1-0-4
Degree $2$
Conductor $260$
Sign $0.348 - 0.937i$
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  + (−0.366 − 1.36i)2-s + (−1.73 + i)4-s + (−1.86 + 1.23i)5-s + (2 + 1.99i)8-s + (−2.59 + 1.5i)9-s + (2.36 + 2.09i)10-s + (0.232 + 3.59i)13-s + (1.99 − 3.46i)16-s + (−1.86 + 6.96i)17-s + (3 + 3i)18-s + (2 − 4i)20-s + (1.96 − 4.59i)25-s + (4.83 − 1.63i)26-s + (−5.76 − 3.33i)29-s + (−5.46 − 1.46i)32-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)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 + (0.748 + 0.663i)10-s + (0.0643 + 0.997i)13-s + (0.499 − 0.866i)16-s + (−0.452 + 1.68i)17-s + (0.707 + 0.707i)18-s + (0.447 − 0.894i)20-s + (0.392 − 0.919i)25-s + (0.947 − 0.320i)26-s + (−1.07 − 0.618i)29-s + (−0.965 − 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.348 - 0.937i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.348 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.425455 + 0.295573i\)
\(L(\frac12)\) \(\approx\) \(0.425455 + 0.295573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
13 \( 1 + (-0.232 - 3.59i)T \)
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 \( 1 + (1.86 - 6.96i)T + (-14.7 - 8.5i)T^{2} \)
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 \( 1 + (0.303 + 1.13i)T + (-32.0 + 18.5i)T^{2} \)
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 \( 1 + (-17.7 - 4.75i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.94026411684814370345053651523, −11.09263525802964340965138972589, −10.68692281218819901896889761040, −9.329945981052943593492469693148, −8.400247076931231053490619797675, −7.60878650336458469676493595936, −6.12655603399409297033607756076, −4.49988109252360874053445407454, −3.53397698297726739596700015736, −2.13845651884361027847611206064, 0.43759024778853351831377871820, 3.37800539749564835381506697996, 4.80979600130612078434710185279, 5.67732661208303335685770569564, 7.01104468523068032294358559032, 7.87783439236222089542648807109, 8.792035243466649634966189173722, 9.476847375838701320946915818045, 10.86426738510035822494080196917, 11.82273598688588294536776995206

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