Properties

Label 2-260-13.12-c1-0-4
Degree $2$
Conductor $260$
Sign $0.998 + 0.0463i$
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  + 2.60·3-s + i·5-s − 2.76i·7-s + 3.76·9-s − 2.16i·11-s + (−0.167 + 3.60i)13-s + 2.60i·15-s + 5.03i·19-s − 7.20i·21-s − 4.93·23-s − 25-s + 2.00·27-s − 4.43·29-s − 3.37i·31-s − 5.63i·33-s + ⋯
L(s)  = 1  + 1.50·3-s + 0.447i·5-s − 1.04i·7-s + 1.25·9-s − 0.653i·11-s + (−0.0463 + 0.998i)13-s + 0.671i·15-s + 1.15i·19-s − 1.57i·21-s − 1.02·23-s − 0.200·25-s + 0.384·27-s − 0.823·29-s − 0.605i·31-s − 0.981i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.998 + 0.0463i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.998 + 0.0463i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91717 - 0.0444379i\)
\(L(\frac12)\) \(\approx\) \(1.91717 - 0.0444379i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (0.167 - 3.60i)T \)
good3 \( 1 - 2.60T + 3T^{2} \)
7 \( 1 + 2.76iT - 7T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.03iT - 19T^{2} \)
23 \( 1 + 4.93T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 + 3.37iT - 31T^{2} \)
37 \( 1 - 3.97iT - 37T^{2} \)
41 \( 1 + 1.66iT - 41T^{2} \)
43 \( 1 - 9.80T + 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 - 8.16iT - 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 7.10iT - 67T^{2} \)
71 \( 1 - 16.1iT - 71T^{2} \)
73 \( 1 + 15.9iT - 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 3.97iT - 83T^{2} \)
89 \( 1 - 7.94iT - 89T^{2} \)
97 \( 1 - 0.462iT - 97T^{2} \)
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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

−12.01059085311433131513018270972, −10.85060556579051053459800402540, −9.939604970373444136465959354114, −9.081145924994609554438741163390, −8.008731783985377883890187855713, −7.37797870919718603160532688693, −6.13715016848265939728381532286, −4.15705663020660915758076541043, −3.44543364396445653262921987706, −1.96234101486075831207888516286, 2.10439963533845728292280067901, 3.12866801358526577020037976333, 4.57530830853088334042073220885, 5.86319560045262701450559114725, 7.46351719007087030448137519390, 8.205474628081237563833072737878, 9.138356089791773384266449276254, 9.613574490610498590131044292874, 10.99662253907658446783162451049, 12.39685749323058580939921505370

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