↑

Properties

Label 2-260-65.4-c1-0-5
Degree $2$
Conductor $260$
Sign $-0.816 + 0.577i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 − 0.739i)3-s + (−0.494 + 2.18i)5-s + (−1.56 − 2.71i)7-s + (−0.406 − 0.704i)9-s + (−4.01 − 2.31i)11-s + (2.44 − 2.64i)13-s + (2.24 − 2.42i)15-s + (−5.87 + 3.38i)17-s + (1.45 − 0.839i)19-s + 4.63i·21-s + (−4.79 − 2.76i)23-s + (−4.51 − 2.15i)25-s + 5.63i·27-s + (3.87 − 6.70i)29-s + 1.46i·31-s + ⋯
L(s)  = 1  + (−0.739 − 0.426i)3-s + (−0.220 + 0.975i)5-s + (−0.592 − 1.02i)7-s + (−0.135 − 0.234i)9-s + (−1.20 − 0.698i)11-s + (0.678 − 0.734i)13-s + (0.579 − 0.626i)15-s + (−1.42 + 0.821i)17-s + (0.333 − 0.192i)19-s + 1.01i·21-s + (−0.999 − 0.576i)23-s + (−0.902 − 0.431i)25-s + 1.08i·27-s + (0.718 − 1.24i)29-s + 0.262i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120579 - 0.379316i\)
\(L(\frac12)\) \(\approx\) \(0.120579 - 0.379316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.494 - 2.18i)T \)
13 \( 1 + (-2.44 + 2.64i)T \)
good3 \( 1 + (1.28 + 0.739i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.56 + 2.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.01 + 2.31i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.87 - 3.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 0.839i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.79 + 2.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.87 + 6.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (3.72 - 6.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.78 - 2.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.7 + 6.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.97T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (-2.27 + 1.31i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.87 + 8.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.317 + 0.550i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.0 - 6.93i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + 3.33T + 83T^{2} \)
89 \( 1 + (2.27 + 1.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.13 + 5.43i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.38745419995054556680105787605, −10.70170715463549809152638354629, −10.18498414830491394449068141088, −8.495017755861849227322351268928, −7.48985101238462481755982959547, −6.45340915742336468263876167452, −5.89562539476740320370215093756, −4.07511792077611965256892768061, −2.86040385028061529440538291027, −0.31923093234832475110704762849, 2.36006812926852517591777606730, 4.31509101992013452664765553438, 5.23018960219933838088951572132, 6.02599859054050054297414557884, 7.50861175744512458768094981476, 8.749497852271526439700982897198, 9.396501582250383552166479236388, 10.56585336681573616149525915123, 11.50927887180204021631988257390, 12.27499893736088241827640558436

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