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Properties

Label 2-1260-1260.299-c0-0-1
Degree $2$
Conductor $1260$
Sign $0.296 - 0.954i$
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  + (−0.866 + 0.5i)2-s + i·3-s + (0.499 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s − 9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)18-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)21-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + i·3-s + (0.499 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s + 0.999i·8-s − 9-s + (−0.866 + 0.5i)10-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)18-s + (0.499 − 0.866i)20-s + (0.5 + 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.296 - 0.954i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.296 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9125312800\)
\(L(\frac12)\) \(\approx\) \(0.9125312800\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 - iT \)
5 \( 1 - T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.940848995127217952003674883753, −9.306651234973315785482553322764, −8.593839071132634729863613641938, −7.77447395491658039612724335167, −6.81107651978977083227175357615, −5.75206331008691630025875432234, −5.24208560702033184641257073411, −4.23524928762237560044066381937, −2.72231303935733839428552139249, −1.47622606909111370400746695295, 1.27775670246214999211020276102, 2.17321762946528768849451510680, 2.92012683967461966255106766222, 4.63177102056873403481277271958, 5.85071292438049271412489111453, 6.48349043539424484280093575736, 7.51332528724776627736805770340, 8.145402703022354266697805541563, 8.975735531125775780091453376363, 9.481475909404073127642105193662

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