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Properties

Label 2-4140-15.2-c1-0-18
Degree $2$
Conductor $4140$
Sign $0.998 + 0.0618i$
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  + (−2.01 − 0.963i)5-s + (2.04 + 2.04i)7-s − 4.66i·11-s + (−0.347 + 0.347i)13-s + (−1.10 + 1.10i)17-s + 4.66i·19-s + (−0.707 − 0.707i)23-s + (3.14 + 3.88i)25-s + 4.98·29-s + 1.77·31-s + (−2.15 − 6.10i)35-s + (−1.84 − 1.84i)37-s − 0.953i·41-s + (−2.26 + 2.26i)43-s + (−2.18 + 2.18i)47-s + ⋯
L(s)  = 1  + (−0.902 − 0.430i)5-s + (0.773 + 0.773i)7-s − 1.40i·11-s + (−0.0963 + 0.0963i)13-s + (−0.267 + 0.267i)17-s + 1.06i·19-s + (−0.147 − 0.147i)23-s + (0.628 + 0.777i)25-s + 0.925·29-s + 0.319·31-s + (−0.364 − 1.03i)35-s + (−0.303 − 0.303i)37-s − 0.148i·41-s + (−0.346 + 0.346i)43-s + (−0.318 + 0.318i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.998 + 0.0618i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.998 + 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.609457867\)
\(L(\frac12)\) \(\approx\) \(1.609457867\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 + 0.963i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-2.04 - 2.04i)T + 7iT^{2} \)
11 \( 1 + 4.66iT - 11T^{2} \)
13 \( 1 + (0.347 - 0.347i)T - 13iT^{2} \)
17 \( 1 + (1.10 - 1.10i)T - 17iT^{2} \)
19 \( 1 - 4.66iT - 19T^{2} \)
29 \( 1 - 4.98T + 29T^{2} \)
31 \( 1 - 1.77T + 31T^{2} \)
37 \( 1 + (1.84 + 1.84i)T + 37iT^{2} \)
41 \( 1 + 0.953iT - 41T^{2} \)
43 \( 1 + (2.26 - 2.26i)T - 43iT^{2} \)
47 \( 1 + (2.18 - 2.18i)T - 47iT^{2} \)
53 \( 1 + (2.24 + 2.24i)T + 53iT^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 3.51T + 61T^{2} \)
67 \( 1 + (0.135 + 0.135i)T + 67iT^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (-4.35 + 4.35i)T - 73iT^{2} \)
79 \( 1 - 8.72iT - 79T^{2} \)
83 \( 1 + (2.66 + 2.66i)T + 83iT^{2} \)
89 \( 1 - 3.24T + 89T^{2} \)
97 \( 1 + (8.18 + 8.18i)T + 97iT^{2} \)
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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

−8.438969293151547768790978350261, −8.001243945031324287771246084069, −7.00739158307580769931133260436, −6.07497453682578095992416019856, −5.43650549635195140484196451908, −4.66496499709153158305273522260, −3.82289415621797637550408911679, −3.04686082151826718154852357567, −1.88695189713261553568767952806, −0.73628883837163230438634707494, 0.71815013697589510127266199827, 2.00496298416040924242024373281, 2.96146610478317752226806511944, 4.02274400232901958725316869282, 4.59970597162615793881061869071, 5.13659829324074273023218733513, 6.61154994998913004996145851110, 6.99859204033475818828538978273, 7.65117049157898958215819534459, 8.231541808241760922359073494844

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