Properties

Label 2-4140-345.344-c1-0-33
Degree $2$
Conductor $4140$
Sign $0.892 + 0.451i$
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  + (1.83 − 1.27i)5-s + 0.920·7-s + 4.88·11-s − 0.822i·13-s − 5.93i·17-s + 7.43i·19-s + (−2.12 + 4.29i)23-s + (1.75 − 4.68i)25-s − 2.63i·29-s + 5.24·31-s + (1.69 − 1.17i)35-s + 7.23·37-s + 1.93i·41-s − 1.42·43-s + 13.2·47-s + ⋯
L(s)  = 1  + (0.821 − 0.569i)5-s + 0.347·7-s + 1.47·11-s − 0.228i·13-s − 1.43i·17-s + 1.70i·19-s + (−0.443 + 0.896i)23-s + (0.350 − 0.936i)25-s − 0.488i·29-s + 0.941·31-s + (0.285 − 0.198i)35-s + 1.18·37-s + 0.301i·41-s − 0.217·43-s + 1.93·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.892 + 0.451i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.737625419\)
\(L(\frac12)\) \(\approx\) \(2.737625419\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 + 1.27i)T \)
23 \( 1 + (2.12 - 4.29i)T \)
good7 \( 1 - 0.920T + 7T^{2} \)
11 \( 1 - 4.88T + 11T^{2} \)
13 \( 1 + 0.822iT - 13T^{2} \)
17 \( 1 + 5.93iT - 17T^{2} \)
19 \( 1 - 7.43iT - 19T^{2} \)
29 \( 1 + 2.63iT - 29T^{2} \)
31 \( 1 - 5.24T + 31T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 - 1.93iT - 41T^{2} \)
43 \( 1 + 1.42T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 - 0.0569iT - 53T^{2} \)
59 \( 1 - 2.50iT - 59T^{2} \)
61 \( 1 + 5.95iT - 61T^{2} \)
67 \( 1 + 5.00T + 67T^{2} \)
71 \( 1 - 8.61iT - 71T^{2} \)
73 \( 1 - 3.83iT - 73T^{2} \)
79 \( 1 + 5.42iT - 79T^{2} \)
83 \( 1 + 1.52iT - 83T^{2} \)
89 \( 1 - 4.74T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−8.354219180102450754319821989032, −7.74814277949993556653051794431, −6.81702081118688174801679326930, −6.02423435218676488651472926058, −5.54056789960198574763621351850, −4.55951126646250756578089691910, −3.92026852459742003408516573035, −2.78155947232956653969468365356, −1.71084829914973424237543259582, −0.966647099727407134971856811196, 1.08536163466488168867513590405, 2.03709836792877624274048113875, 2.88606468669575940410042172334, 3.99333120078959157660777337827, 4.60376522450170138784665292797, 5.67304016109160303779526789752, 6.47492683578729388847384645083, 6.69508825839481337554503824451, 7.70240890275167875469070480088, 8.680316249219364867725362582842

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