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Properties

Label 2-4140-15.8-c1-0-6
Degree $2$
Conductor $4140$
Sign $-0.995 + 0.0897i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 + 1.01i)5-s + (−1.87 + 1.87i)7-s + 1.74i·11-s + (3.99 + 3.99i)13-s + (−0.582 − 0.582i)17-s + 7.02i·19-s + (0.707 − 0.707i)23-s + (2.92 − 4.05i)25-s − 4.66·29-s + 2.39·31-s + (1.82 − 5.64i)35-s + (−7.42 + 7.42i)37-s + 5.25i·41-s + (−4.98 − 4.98i)43-s + (7.04 + 7.04i)47-s + ⋯
L(s)  = 1  + (−0.889 + 0.456i)5-s + (−0.709 + 0.709i)7-s + 0.524i·11-s + (1.10 + 1.10i)13-s + (−0.141 − 0.141i)17-s + 1.61i·19-s + (0.147 − 0.147i)23-s + (0.584 − 0.811i)25-s − 0.865·29-s + 0.430·31-s + (0.307 − 0.954i)35-s + (−1.22 + 1.22i)37-s + 0.820i·41-s + (−0.760 − 0.760i)43-s + (1.02 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.995 + 0.0897i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.995 + 0.0897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8155253427\)
\(L(\frac12)\) \(\approx\) \(0.8155253427\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1.99 - 1.01i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (1.87 - 1.87i)T - 7iT^{2} \)
11 \( 1 - 1.74iT - 11T^{2} \)
13 \( 1 + (-3.99 - 3.99i)T + 13iT^{2} \)
17 \( 1 + (0.582 + 0.582i)T + 17iT^{2} \)
19 \( 1 - 7.02iT - 19T^{2} \)
29 \( 1 + 4.66T + 29T^{2} \)
31 \( 1 - 2.39T + 31T^{2} \)
37 \( 1 + (7.42 - 7.42i)T - 37iT^{2} \)
41 \( 1 - 5.25iT - 41T^{2} \)
43 \( 1 + (4.98 + 4.98i)T + 43iT^{2} \)
47 \( 1 + (-7.04 - 7.04i)T + 47iT^{2} \)
53 \( 1 + (-7.72 + 7.72i)T - 53iT^{2} \)
59 \( 1 - 7.38T + 59T^{2} \)
61 \( 1 - 0.462T + 61T^{2} \)
67 \( 1 + (-5.30 + 5.30i)T - 67iT^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-10.1 - 10.1i)T + 73iT^{2} \)
79 \( 1 - 4.66iT - 79T^{2} \)
83 \( 1 + (-1.85 + 1.85i)T - 83iT^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (10.6 - 10.6i)T - 97iT^{2} \)
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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

−8.584896314534057326224349457717, −8.247222680001318556586726471070, −7.21009974649381550830329471317, −6.62382814896163799304037506729, −6.03182496934760141104145373280, −5.06024955917226501072602798558, −3.96682087588529117008485892604, −3.60386768406519287601837518654, −2.54930168106029874514037158840, −1.48893314866031416128353894166, 0.28321528024924465733386257627, 1.00648714685400548644157783338, 2.65107462000334622234067333573, 3.67435222343956011944581786613, 3.88897534701387100286910313865, 5.12248454905119315544064441203, 5.71252833872787431790811165136, 6.81490123069730284898439593372, 7.21384594445954527524505505191, 8.127282880217872088953936271451

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