Properties

Label 2-260-260.23-c1-0-26
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: $\chi_{260} (23, \cdot )$
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

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

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