Properties

Label 2-4140-1.1-c1-0-18
Degree $2$
Conductor $4140$
Sign $1$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 1.52·7-s + 3.59·11-s + 5.59·13-s − 4.07·17-s − 1.59·19-s − 23-s + 25-s + 4.07·29-s + 2.47·31-s + 1.52·35-s + 9.66·37-s − 5.01·41-s + 7.05·43-s + 4.54·47-s − 4.66·49-s − 5.01·53-s + 3.59·55-s − 4.07·59-s + 6.54·61-s + 5.59·65-s + 2.47·67-s + 5.01·71-s − 8.79·73-s + 5.49·77-s + 2.94·79-s − 9.12·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.576·7-s + 1.08·11-s + 1.55·13-s − 0.987·17-s − 0.366·19-s − 0.208·23-s + 0.200·25-s + 0.756·29-s + 0.444·31-s + 0.258·35-s + 1.58·37-s − 0.783·41-s + 1.07·43-s + 0.663·47-s − 0.667·49-s − 0.689·53-s + 0.485·55-s − 0.530·59-s + 0.838·61-s + 0.694·65-s + 0.302·67-s + 0.595·71-s − 1.02·73-s + 0.625·77-s + 0.331·79-s − 1.00·83-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.667470231\)
\(L(\frac12)\) \(\approx\) \(2.667470231\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 - 1.52T + 7T^{2} \)
11 \( 1 - 3.59T + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 + 4.07T + 17T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 - 9.66T + 37T^{2} \)
41 \( 1 + 5.01T + 41T^{2} \)
43 \( 1 - 7.05T + 43T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 + 5.01T + 53T^{2} \)
59 \( 1 + 4.07T + 59T^{2} \)
61 \( 1 - 6.54T + 61T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 - 5.01T + 71T^{2} \)
73 \( 1 + 8.79T + 73T^{2} \)
79 \( 1 - 2.94T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−8.537581665840920820538938978246, −7.82579156835232422114346742315, −6.67183457801333938463394652085, −6.35328800989069438960007878873, −5.57628269647893686768030019779, −4.45898978061860136248685448506, −4.03814060687521547228709071863, −2.89498398684534227420128739029, −1.82678201657040856791767364805, −1.01291566206120024120795033286, 1.01291566206120024120795033286, 1.82678201657040856791767364805, 2.89498398684534227420128739029, 4.03814060687521547228709071863, 4.45898978061860136248685448506, 5.57628269647893686768030019779, 6.35328800989069438960007878873, 6.67183457801333938463394652085, 7.82579156835232422114346742315, 8.537581665840920820538938978246

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