↑

Properties

Label 2-260-260.103-c1-0-18
Degree $2$
Conductor $260$
Sign $0.778 + 0.627i$
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 − i)2-s + 2i·4-s + (2 − i)5-s + (2 − 2i)8-s + 3i·9-s + (−3 − i)10-s + (3 + 2i)13-s − 4·16-s + (5 − 5i)17-s + (3 − 3i)18-s + (2 + 4i)20-s + (3 − 4i)25-s + (−1 − 5i)26-s − 4i·29-s + (4 + 4i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·4-s + (0.894 − 0.447i)5-s + (0.707 − 0.707i)8-s + i·9-s + (−0.948 − 0.316i)10-s + (0.832 + 0.554i)13-s − 16-s + (1.21 − 1.21i)17-s + (0.707 − 0.707i)18-s + (0.447 + 0.894i)20-s + (0.600 − 0.800i)25-s + (−0.196 − 0.980i)26-s − 0.742i·29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00719 - 0.355110i\)
\(L(\frac12)\) \(\approx\) \(1.00719 - 0.355110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (-2 + i)T \)
13 \( 1 + (-3 - 2i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (7 + 7i)T + 37iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (11 - 11i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (13 + 13i)T + 97iT^{2} \)
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

−11.74738049845887365154618954305, −10.82796214410643999612947613927, −9.941594532608218145202032347927, −9.187547350294419413295940592773, −8.229911823665325104164598562011, −7.20957539847906622450927133130, −5.73491143410523833211528396224, −4.47535795302851131679368478569, −2.80530829135999873954974062025, −1.45691168177840979857636693211, 1.45561063744892437789702457079, 3.45356325176892242942731559183, 5.43462547299117521541803966787, 6.15673848355926331452691721590, 7.04059020109515581182621511409, 8.323425093951738756602128044480, 9.131908422051712165194823091286, 10.19697333540397973062953186803, 10.63023964168841980145049918380, 12.03544996128417207156173930919

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