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Properties

Label 2-2420-220.119-c0-0-15
Degree $2$
Conductor $2420$
Sign $0.0219 - 0.999i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−1.61 − 1.17i)7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)9-s − 0.999·10-s + (−1.61 + 1.17i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.309 + 0.951i)20-s + (−0.809 + 0.587i)25-s + (0.618 + 1.90i)28-s + 32-s + (−0.618 + 1.90i)35-s + (0.809 − 0.587i)36-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−1.61 − 1.17i)7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)9-s − 0.999·10-s + (−1.61 + 1.17i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)18-s + (−0.309 + 0.951i)20-s + (−0.809 + 0.587i)25-s + (0.618 + 1.90i)28-s + 32-s + (−0.618 + 1.90i)35-s + (0.809 − 0.587i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(1.20773\)
Root analytic conductor: \(1.09897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2420} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2420,\ (\ :0),\ 0.0219 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2001735845\)
\(L(\frac12)\) \(\approx\) \(0.2001735845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.732374154359388665989345279625, −7.964497424312869054983022379769, −7.01322466536189607229251238172, −6.05126102420933720144609526415, −5.16706538640018035616187599851, −4.37480340267060552187867163191, −3.64733262421566775585584694051, −2.83200716169715807325540969555, −1.47081152636761164062523681828, −0.12037609521046361428675148153, 2.67997851464237829048874250766, 3.28063194379875863792711471552, 3.94641728213048282334528627064, 5.35430594158067234256476488945, 6.04371309032712522022347173618, 6.62416055803054729641269262875, 7.02798661879323559499529582402, 8.215412207932390924843772809882, 8.832406935530036220953587849836, 9.644788669216947498795552593217

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