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Properties

Label 2-260-260.43-c1-0-25
Degree $2$
Conductor $260$
Sign $0.990 + 0.141i$
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s + (−0.133 + 2.23i)5-s + (1.99 − 2i)8-s + (2.59 − 1.5i)9-s + (0.633 + 3.09i)10-s + (−3.23 + 1.59i)13-s + (1.99 − 3.46i)16-s + (−0.133 − 0.0358i)17-s + (3 − 3i)18-s + (2 + 3.99i)20-s + (−4.96 − 0.598i)25-s + (−3.83 + 3.36i)26-s + (−9.23 − 5.33i)29-s + (1.46 − 5.46i)32-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + (−0.0599 + 0.998i)5-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (0.200 + 0.979i)10-s + (−0.896 + 0.443i)13-s + (0.499 − 0.866i)16-s + (−0.0324 − 0.00870i)17-s + (0.707 − 0.707i)18-s + (0.447 + 0.894i)20-s + (−0.992 − 0.119i)25-s + (−0.751 + 0.660i)26-s + (−1.71 − 0.989i)29-s + (0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.990 + 0.141i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.990 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19611 - 0.155642i\)
\(L(\frac12)\) \(\approx\) \(2.19611 - 0.155642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 + (0.133 - 2.23i)T \)
13 \( 1 + (3.23 - 1.59i)T \)
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 \( 1 + (0.133 + 0.0358i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (9.23 + 5.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (10.6 - 2.86i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-10.3 - 5.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-5.29 - 5.29i)T + 53iT^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.16 + 1.16i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.75 - 17.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.98021849873678474732146875670, −11.22008427892027977458233359061, −10.21471240296305693964805836261, −9.509050851451376614445355339151, −7.55302108494919958138867422268, −6.91142190369548993287531999010, −5.89933104727720344581735926307, −4.49210578042835974567409388455, −3.46000644685835524707585614927, −2.08898932961594158668462306208, 1.95942638812661455191096502919, 3.76036702768594012231891319056, 4.86561018704451972422094923930, 5.56958272668486288646026675146, 7.11521259470731472799974629183, 7.79550374872632366999366220395, 9.040700637076031427621710515298, 10.25188009445651792846460030184, 11.28209905854596109124676965112, 12.54160495714783284555854044482

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