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Properties

Label 2-1260-105.74-c0-0-0
Degree $2$
Conductor $1260$
Sign $-0.286 - 0.958i$
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  + (−0.965 − 0.258i)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 + (0.965 − 0.258i)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  + (−0.965 − 0.258i)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 + (0.965 − 0.258i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.286 - 0.958i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5421548654\)
\(L(\frac12)\) \(\approx\) \(0.5421548654\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
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

−10.23409557431517017649579814996, −8.950399568749888273989848067272, −8.778765446423895073357804994128, −7.71026971603166349931988538095, −6.77219149641461020901166922279, −6.13629513688679734349644112607, −4.94556141685006509697200062546, −3.99282451379923996282847357908, −3.24497150741637261208656323365, −1.79428896087623349626107626926, 0.45689532563816288200387315070, 2.62346757628067691120612645287, 3.48100328327064860352772278580, 4.32670325324356331419844706423, 5.43109437496626043576637940879, 6.45649862515445015640148528063, 7.42048225685788720632866918721, 7.68472300837040210079533583376, 8.926413044468902832014422924835, 9.652608572437304468780350723530

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