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Properties

Label 2-2420-55.29-c0-0-0
Degree $2$
Conductor $2420$
Sign $0.520 + 0.853i$
Analytic cond. $1.20773$
Root an. cond. $1.09897$
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  + (−1.64 − 0.535i)3-s + (−0.913 − 0.406i)5-s + (1.61 + 1.17i)9-s + (1.28 + 1.15i)15-s + 1.73i·23-s + (0.669 + 0.743i)25-s + (−1.01 − 1.40i)27-s + (−0.809 − 0.587i)31-s + (1.64 − 0.535i)37-s + (−1 − 1.73i)45-s + (0.809 − 0.587i)49-s + (−0.309 − 0.951i)59-s − 1.73i·67-s + (0.927 − 2.85i)69-s + (0.809 − 0.587i)71-s + ⋯
L(s)  = 1  + (−1.64 − 0.535i)3-s + (−0.913 − 0.406i)5-s + (1.61 + 1.17i)9-s + (1.28 + 1.15i)15-s + 1.73i·23-s + (0.669 + 0.743i)25-s + (−1.01 − 1.40i)27-s + (−0.809 − 0.587i)31-s + (1.64 − 0.535i)37-s + (−1 − 1.73i)45-s + (0.809 − 0.587i)49-s + (−0.309 − 0.951i)59-s − 1.73i·67-s + (0.927 − 2.85i)69-s + (0.809 − 0.587i)71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(1.20773\)
Root analytic conductor: \(1.09897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2420} (1129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2420,\ (\ :0),\ 0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4962818801\)
\(L(\frac12)\) \(\approx\) \(0.4962818801\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 \)
good3 \( 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
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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

−9.077900533323415877295730364182, −7.83235862049387657851605470427, −7.53124156960624494584949196104, −6.66171271921706895787138062832, −5.82387787953380248747902203385, −5.21773977709768336674322987542, −4.41273852225584705310534834890, −3.50283287176845941773047454822, −1.81259670236097633496321265357, −0.62817984632745541366758465576, 0.841891944137098416841549083779, 2.71272961094953116670570909071, 3.97243721129919209701194108004, 4.47761276051620588849064946963, 5.30728610234815652496287145561, 6.20761805133331346273157191426, 6.75107323328392659622594022778, 7.55191471308409077477552298973, 8.485610388831270012493366963338, 9.416309511493353418302307143936

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