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Properties

Label 2-260-260.23-c1-0-18
Degree $2$
Conductor $260$
Sign $0.995 + 0.0918i$
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  + (1.30 − 0.554i)2-s + (0.754 + 0.202i)3-s + (1.38 − 1.44i)4-s + (0.282 + 2.21i)5-s + (1.09 − 0.155i)6-s + (0.814 + 3.04i)7-s + (1.00 − 2.64i)8-s + (−2.06 − 1.19i)9-s + (1.59 + 2.72i)10-s + (0.846 + 1.46i)11-s + (1.33 − 0.809i)12-s + (−2.00 − 2.99i)13-s + (2.74 + 3.50i)14-s + (−0.235 + 1.73i)15-s + (−0.164 − 3.99i)16-s + (−1.49 − 5.58i)17-s + ⋯
L(s)  = 1  + (0.919 − 0.392i)2-s + (0.435 + 0.116i)3-s + (0.692 − 0.721i)4-s + (0.126 + 0.991i)5-s + (0.446 − 0.0634i)6-s + (0.308 + 1.14i)7-s + (0.353 − 0.935i)8-s + (−0.689 − 0.398i)9-s + (0.505 + 0.862i)10-s + (0.255 + 0.442i)11-s + (0.385 − 0.233i)12-s + (−0.555 − 0.831i)13-s + (0.734 + 0.936i)14-s + (−0.0607 + 0.446i)15-s + (−0.0412 − 0.999i)16-s + (−0.363 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.995 + 0.0918i$
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: \((2,\ 260,\ (\ :1/2),\ 0.995 + 0.0918i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33169 - 0.107326i\)
\(L(\frac12)\) \(\approx\) \(2.33169 - 0.107326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.554i)T \)
5 \( 1 + (-0.282 - 2.21i)T \)
13 \( 1 + (2.00 + 2.99i)T \)
good3 \( 1 + (-0.754 - 0.202i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.814 - 3.04i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.846 - 1.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.49 + 5.58i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.47 + 0.853i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.69 + 0.453i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.31 + 0.758i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.65T + 31T^{2} \)
37 \( 1 + (0.693 - 2.58i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.12 - 0.647i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.38 - 8.89i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (7.31 - 7.31i)T - 47iT^{2} \)
53 \( 1 + (-4.69 - 4.69i)T + 53iT^{2} \)
59 \( 1 + (5.38 + 3.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.75 + 3.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.9 - 3.73i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.41 + 4.18i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.10 - 8.10i)T - 73iT^{2} \)
79 \( 1 + 9.28T + 79T^{2} \)
83 \( 1 + (7.11 + 7.11i)T + 83iT^{2} \)
89 \( 1 + (-0.710 - 1.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.76 - 1.00i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.80217802619573937023125764236, −11.42851057478709583166746468969, −10.12812717359535948008644368659, −9.370669593349438372848238293824, −8.006233346357430418455898130152, −6.71138647192624059248183181645, −5.81746006530508101763394624018, −4.66201302982061784524926720805, −2.98699318766297150120040079925, −2.49307551417460960088808180056, 1.97146710983620546624648273573, 3.79211609705481500379041787159, 4.61948255464203880159135618516, 5.81970873703501982611836487046, 6.98353235279797191297614513094, 8.158072678505456476068271516503, 8.648574668693555479310364853499, 10.23749309707011942422290207627, 11.31782042130870767481879084433, 12.18637557354871792265667856109

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