↑

Properties

Label 2-260-65.58-c1-0-1
Degree $2$
Conductor $260$
Sign $0.686 + 0.727i$
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  + (−3.27 + 0.877i)3-s + (−1.78 + 1.34i)5-s + (−1.31 − 2.27i)7-s + (7.34 − 4.24i)9-s + (1.71 − 0.458i)11-s + (3.39 + 1.21i)13-s + (4.66 − 5.97i)15-s + (0.0972 − 0.363i)17-s + (0.462 − 1.72i)19-s + (6.28 + 6.28i)21-s + (−1.26 − 4.72i)23-s + (1.37 − 4.80i)25-s + (−13.1 + 13.1i)27-s + (−6.28 − 3.62i)29-s + (2.98 − 2.98i)31-s + ⋯
L(s)  = 1  + (−1.88 + 0.506i)3-s + (−0.798 + 0.601i)5-s + (−0.495 − 0.859i)7-s + (2.44 − 1.41i)9-s + (0.516 − 0.138i)11-s + (0.941 + 0.337i)13-s + (1.20 − 1.54i)15-s + (0.0235 − 0.0880i)17-s + (0.106 − 0.395i)19-s + (1.37 + 1.37i)21-s + (−0.264 − 0.986i)23-s + (0.275 − 0.961i)25-s + (−2.52 + 2.52i)27-s + (−1.16 − 0.674i)29-s + (0.535 − 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445958 - 0.192311i\)
\(L(\frac12)\) \(\approx\) \(0.445958 - 0.192311i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.78 - 1.34i)T \)
13 \( 1 + (-3.39 - 1.21i)T \)
good3 \( 1 + (3.27 - 0.877i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.31 + 2.27i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.71 + 0.458i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.0972 + 0.363i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.462 + 1.72i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.26 + 4.72i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.28 + 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.98 + 2.98i)T - 31iT^{2} \)
37 \( 1 + (-5.10 + 8.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.59 - 5.96i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.59 - 0.427i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 + (-5.13 - 5.13i)T + 53iT^{2} \)
59 \( 1 + (-2.89 - 0.775i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.45 + 4.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.90 + 3.98i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.02 - 1.07i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 2.72iT - 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + (1.91 + 7.14i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.45 - 0.838i)T + (48.5 - 84.0i)T^{2} \)
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

−11.54149758534716253807757515120, −11.05084365533865521472704414635, −10.35408949462713420709823797020, −9.342202136443130876765074826767, −7.55591201689010072099901765981, −6.60991730477404586883066756673, −6.00300083027332633571525871912, −4.40210707940155201000695731849, −3.81051878589100938134940727871, −0.56962147025493037448880781318, 1.27349812155820147592655897038, 3.90854368034734215600158542768, 5.23654460979977281625517741487, 5.93742985908437998841760476224, 6.93685356441970785140713135508, 8.034962758040172452477998719530, 9.322947147753750786976095051640, 10.53489116088482539186413662308, 11.52160065747408037873051549140, 11.94729543747994767098291811528

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