Properties

Label 2-1260-105.44-c0-0-0
Degree $2$
Conductor $1260$
Sign $0.589 - 0.807i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)5-s + (0.866 + 0.5i)7-s + i·13-s + (−0.707 − 1.22i)17-s + (0.5 − 0.866i)19-s + (−0.707 + 1.22i)23-s + (−0.866 + 0.499i)25-s + 1.41i·29-s + (−0.5 − 0.866i)31-s + (−0.258 + 0.965i)35-s + (0.866 + 0.5i)37-s i·43-s + (0.707 − 1.22i)47-s + (0.499 + 0.866i)49-s + (1.22 − 0.707i)59-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)5-s + (0.866 + 0.5i)7-s + i·13-s + (−0.707 − 1.22i)17-s + (0.5 − 0.866i)19-s + (−0.707 + 1.22i)23-s + (−0.866 + 0.499i)25-s + 1.41i·29-s + (−0.5 − 0.866i)31-s + (−0.258 + 0.965i)35-s + (0.866 + 0.5i)37-s i·43-s + (0.707 − 1.22i)47-s + (0.499 + 0.866i)49-s + (1.22 − 0.707i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.589 - 0.807i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.589 - 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.180141928\)
\(L(\frac12)\) \(\approx\) \(1.180141928\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.862334709368663492440585393406, −9.285693731086643752131097993077, −8.464451392522113056610540676952, −7.25571703211120740302604367935, −6.99312325892684930712732122428, −5.76809290454116784545578542467, −5.02548302388329897739331742829, −3.92437692347078370072152167570, −2.70738857401396053801765080566, −1.85144213327307794212628730623, 1.14334442285070001189928614358, 2.31027967555264573545382235650, 3.94140478257495277929620362253, 4.54263608142498219794871128063, 5.59334389371029732370988164765, 6.22805614219987303399182612914, 7.62857318077301701013954443575, 8.156970848363330293232741458941, 8.763826448036795803178881689950, 9.883955658274724994413283605386

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