Properties

Label 2-4140-1.1-c1-0-15
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 + 4.20·7-s − 2.75·11-s − 0.753·13-s + 4.96·17-s + 4.75·19-s − 23-s + 25-s − 4.96·29-s − 0.209·31-s + 4.20·35-s − 5.71·37-s + 9.38·41-s + 12.4·43-s − 7.17·47-s + 10.7·49-s + 9.38·53-s − 2.75·55-s + 4.96·59-s − 5.17·61-s − 0.753·65-s − 0.209·67-s − 9.38·71-s + 10.2·73-s − 11.5·77-s − 2.41·79-s − 5.45·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.59·7-s − 0.830·11-s − 0.208·13-s + 1.20·17-s + 1.09·19-s − 0.208·23-s + 0.200·25-s − 0.921·29-s − 0.0375·31-s + 0.711·35-s − 0.939·37-s + 1.46·41-s + 1.89·43-s − 1.04·47-s + 1.53·49-s + 1.28·53-s − 0.371·55-s + 0.646·59-s − 0.662·61-s − 0.0934·65-s − 0.0255·67-s − 1.11·71-s + 1.20·73-s − 1.32·77-s − 0.272·79-s − 0.598·83-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

\(L(1)\) \(\approx\) \(2.647591692\)
\(L(\frac12)\) \(\approx\) \(2.647591692\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.20T + 7T^{2} \)
11 \( 1 + 2.75T + 11T^{2} \)
13 \( 1 + 0.753T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
29 \( 1 + 4.96T + 29T^{2} \)
31 \( 1 + 0.209T + 31T^{2} \)
37 \( 1 + 5.71T + 37T^{2} \)
41 \( 1 - 9.38T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 - 9.38T + 53T^{2} \)
59 \( 1 - 4.96T + 59T^{2} \)
61 \( 1 + 5.17T + 61T^{2} \)
67 \( 1 + 0.209T + 67T^{2} \)
71 \( 1 + 9.38T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 2.41T + 79T^{2} \)
83 \( 1 + 5.45T + 83T^{2} \)
89 \( 1 + 4.91T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−8.313237247913776491781628023070, −7.52791916103398432197664279856, −7.40011767727810981457358770620, −5.91259818132581918029414095652, −5.42435831907342898944353143969, −4.86598445669859623626863531984, −3.88790489979371730804556037579, −2.80429404863676946273072101567, −1.91798839493946426912604372779, −0.986783123188129510261269133541, 0.986783123188129510261269133541, 1.91798839493946426912604372779, 2.80429404863676946273072101567, 3.88790489979371730804556037579, 4.86598445669859623626863531984, 5.42435831907342898944353143969, 5.91259818132581918029414095652, 7.40011767727810981457358770620, 7.52791916103398432197664279856, 8.313237247913776491781628023070

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