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Properties

Label 2-1260-140.27-c0-0-0
Degree $2$
Conductor $1260$
Sign $0.229 - 0.973i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s − 5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 − 0.707i)10-s + 1.41i·11-s + 1.00·14-s − 1.00·16-s + (1 − i)17-s + 1.41i·19-s + 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (−0.707 + 0.707i)28-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s − 5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 − 0.707i)10-s + 1.41i·11-s + 1.00·14-s − 1.00·16-s + (1 − i)17-s + 1.41i·19-s + 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (−0.707 + 0.707i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5440687215\)
\(L(\frac12)\) \(\approx\) \(0.5440687215\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.904766993883489930405597721328, −9.414520003943571883918605135749, −8.000620581584515570504974511315, −7.80485154690366904561136707100, −6.92705948529366596828024634657, −6.25187600918607182909362750275, −4.93527894313322896603873536411, −4.23742517630422112422632254882, −2.96871323862178985549805258401, −1.18390082080493178738121034416, 0.72225574791084237986371816321, 2.60559888191821740489955136368, 3.34129673297647019853988595519, 4.15959195907902314390785598791, 5.55295642832733716384564206493, 6.57322563157934108168449731210, 7.50601792655953356029545864938, 8.428695319287882281522076442528, 8.753608224797781494274252034240, 9.678022517458487482528824111544

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