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Properties

Label 2-4140-5.4-c1-0-17
Degree $2$
Conductor $4140$
Sign $-0.983 - 0.181i$
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  + (−0.405 + 2.19i)5-s + 3.23i·7-s + 1.54·11-s + 6.47i·13-s + 7.55i·17-s + 1.20·19-s + i·23-s + (−4.67 − 1.78i)25-s + 1.14·29-s − 6.97·31-s + (−7.10 − 1.31i)35-s + 5.33i·37-s + 7.35·41-s + 3.63i·43-s − 10.3i·47-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)5-s + 1.22i·7-s + 0.464·11-s + 1.79i·13-s + 1.83i·17-s + 0.277·19-s + 0.208i·23-s + (−0.934 − 0.356i)25-s + 0.211·29-s − 1.25·31-s + (−1.20 − 0.221i)35-s + 0.877i·37-s + 1.14·41-s + 0.554i·43-s − 1.50i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.983 - 0.181i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.983 - 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.610675030\)
\(L(\frac12)\) \(\approx\) \(1.610675030\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.405 - 2.19i)T \)
23 \( 1 - iT \)
good7 \( 1 - 3.23iT - 7T^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 - 6.47iT - 13T^{2} \)
17 \( 1 - 7.55iT - 17T^{2} \)
19 \( 1 - 1.20T + 19T^{2} \)
29 \( 1 - 1.14T + 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 - 5.33iT - 37T^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 - 3.63iT - 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 3.16iT - 53T^{2} \)
59 \( 1 - 8.15T + 59T^{2} \)
61 \( 1 - 0.160T + 61T^{2} \)
67 \( 1 + 15.1iT - 67T^{2} \)
71 \( 1 - 3.24T + 71T^{2} \)
73 \( 1 + 9.51iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 3.26iT - 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + 9.21iT - 97T^{2} \)
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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

−8.872989383164409264593817299063, −8.051533148653196585923443281873, −7.23470705638931317577581981751, −6.32023612803338911279219611137, −6.18812998334913037474845765414, −5.07705953879800500778526527771, −4.01874362768357286045671369852, −3.47422371877217890140562727036, −2.26989075125737377128920439161, −1.74006149647777357861298374175, 0.53683687832155513978115138460, 1.02108185001736288278435560201, 2.55640106304028548719251033346, 3.58005386985303398275008998149, 4.23715758469268237149434553739, 5.16483531381594324791138482859, 5.58534342520136672793057461824, 6.77445153124924127909779610060, 7.60062098169137465578135988590, 7.79484677334856072527561612582

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