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Properties

Label 2-1260-1260.79-c0-0-0
Degree $2$
Conductor $1260$
Sign $-0.400 - 0.916i$
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  + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + 9-s + (0.5 + 0.866i)10-s + (0.499 − 0.866i)12-s + (−0.499 + 0.866i)14-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (−0.499 + 0.866i)20-s + (−0.5 − 0.866i)21-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + 9-s + (0.5 + 0.866i)10-s + (0.499 − 0.866i)12-s + (−0.499 + 0.866i)14-s − 15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)18-s + (−0.499 + 0.866i)20-s + (−0.5 − 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158482180\)
\(L(\frac12)\) \(\approx\) \(1.158482180\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 - T + T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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

−10.10565276950189096421681690271, −9.125033515723090958650605778923, −8.613625797559795280706455998796, −7.30330046935198458234703319073, −6.73110307643248338198472121730, −5.75331704399695372293962118306, −5.34260740557641570116647952362, −4.66054248793872953131305007481, −3.23488698070796818721714174924, −1.78854771127628967216023435466, 1.08066306694577311942223332024, 2.10277314183847177237462616857, 3.57193518369719423967751122731, 4.65874982858307898869923180944, 5.18148788038742473475327454338, 6.14124722348907347234077433588, 6.79041848317970311452044166939, 7.985637042320783740292450537461, 9.324244686142360333497034116317, 9.877842846919813235236943117042

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