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Properties

Label 2-2420-55.24-c0-0-1
Degree $2$
Conductor $2420$
Sign $-0.162 + 0.986i$
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  + (−1.01 − 1.40i)3-s + (0.978 + 0.207i)5-s + (−0.618 + 1.90i)9-s + (−0.704 − 1.58i)15-s − 1.73i·23-s + (0.913 + 0.406i)25-s + (1.64 − 0.535i)27-s + (0.309 − 0.951i)31-s + (1.01 − 1.40i)37-s + (−1 + 1.73i)45-s + (−0.309 − 0.951i)49-s + (0.809 + 0.587i)59-s + 1.73i·67-s + (−2.42 + 1.76i)69-s + (−0.309 − 0.951i)71-s + ⋯
L(s)  = 1  + (−1.01 − 1.40i)3-s + (0.978 + 0.207i)5-s + (−0.618 + 1.90i)9-s + (−0.704 − 1.58i)15-s − 1.73i·23-s + (0.913 + 0.406i)25-s + (1.64 − 0.535i)27-s + (0.309 − 0.951i)31-s + (1.01 − 1.40i)37-s + (−1 + 1.73i)45-s + (−0.309 − 0.951i)49-s + (0.809 + 0.587i)59-s + 1.73i·67-s + (−2.42 + 1.76i)69-s + (−0.309 − 0.951i)71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-0.162 + 0.986i$
Analytic conductor: \(1.20773\)
Root analytic conductor: \(1.09897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2420} (2169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2420,\ (\ :0),\ -0.162 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9605098886\)
\(L(\frac12)\) \(\approx\) \(0.9605098886\)
\(L(1)\) 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 \( 1 + (-0.978 - 0.207i)T \)
11 \( 1 \)
good3 \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.73iT - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - 1.73iT - T^{2} \)
71 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2} \)
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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

−8.814022833592892713830938172846, −7.994749376415413923448931348137, −7.14289614903849987035939741330, −6.56786780925512996443611790917, −5.94193018893473635246548193971, −5.36857669331909591070668340153, −4.32350268515272701878975756253, −2.65301150462523256419588941258, −2.00261291112293776846983326010, −0.807548678767695024771952773920, 1.35555873613772083650341355257, 2.90887961881098841154525553775, 3.86894818544094382151212380046, 4.85035981630868936256056722047, 5.28438275483022837769093153364, 6.06338986766385028021853324888, 6.67160561977006704313795848801, 7.916858411802860480791844515618, 8.983961498476212403669743618744, 9.539468269143441542457580183589

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