Properties

Label 2-260-65.57-c1-0-1
Degree $2$
Conductor $260$
Sign $0.163 - 0.986i$
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  + (−0.285 + 0.285i)3-s + (−2.21 − 0.285i)5-s + 1.26i·7-s + 2.83i·9-s + (4.21 + 4.21i)11-s + (2.28 + 2.78i)13-s + (0.714 − 0.551i)15-s + (−1.93 + 1.93i)17-s + (−2.55 − 2.55i)19-s + (−0.361 − 0.361i)21-s + (−5.21 − 5.21i)23-s + (4.83 + 1.26i)25-s + (−1.66 − 1.66i)27-s + 7.16i·29-s + (2.78 − 2.78i)31-s + ⋯
L(s)  = 1  + (−0.164 + 0.164i)3-s + (−0.991 − 0.127i)5-s + 0.478i·7-s + 0.945i·9-s + (1.27 + 1.27i)11-s + (0.633 + 0.773i)13-s + (0.184 − 0.142i)15-s + (−0.468 + 0.468i)17-s + (−0.585 − 0.585i)19-s + (−0.0788 − 0.0788i)21-s + (−1.08 − 1.08i)23-s + (0.967 + 0.253i)25-s + (−0.320 − 0.320i)27-s + 1.33i·29-s + (0.500 − 0.500i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736220 + 0.623968i\)
\(L(\frac12)\) \(\approx\) \(0.736220 + 0.623968i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.285i)T \)
13 \( 1 + (-2.28 - 2.78i)T \)
good3 \( 1 + (0.285 - 0.285i)T - 3iT^{2} \)
7 \( 1 - 1.26iT - 7T^{2} \)
11 \( 1 + (-4.21 - 4.21i)T + 11iT^{2} \)
17 \( 1 + (1.93 - 1.93i)T - 17iT^{2} \)
19 \( 1 + (2.55 + 2.55i)T + 19iT^{2} \)
23 \( 1 + (5.21 + 5.21i)T + 23iT^{2} \)
29 \( 1 - 7.16iT - 29T^{2} \)
31 \( 1 + (-2.78 + 2.78i)T - 31iT^{2} \)
37 \( 1 + 5.83iT - 37T^{2} \)
41 \( 1 + (1.33 - 1.33i)T - 41iT^{2} \)
43 \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (9.27 - 9.27i)T - 53iT^{2} \)
59 \( 1 + (2.55 - 2.55i)T - 59iT^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 1.30T + 67T^{2} \)
71 \( 1 + (-8.72 + 8.72i)T - 71iT^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 + 0.954iT - 79T^{2} \)
83 \( 1 + 5.13iT - 83T^{2} \)
89 \( 1 + (-5.31 + 5.31i)T - 89iT^{2} \)
97 \( 1 - 2.81T + 97T^{2} \)
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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

−12.12519918824875733789768917594, −11.32780252147352495731498146897, −10.47965599513170091440548959237, −9.153688250507355315354620678889, −8.441399960816513283099622492735, −7.24713728296935633023103456727, −6.30889271193634872779361729522, −4.65951416293981164169589773253, −4.05633943027093866350377189028, −2.04862407886934408911503799930, 0.823465290794851858321158793568, 3.43854011658717198517641913230, 4.06172929195261134720903881447, 5.93618945405287464998060816422, 6.69400073078481936632265850278, 7.943167055689186402874576514981, 8.716629684229954353691156277273, 9.895602508946117859065306891032, 11.17218746068743401250195486713, 11.62280670085839796416762556167

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