↑

Properties

Label 2-4140-15.2-c1-0-12
Degree $2$
Conductor $4140$
Sign $0.522 - 0.852i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 + 1.33i)5-s + (−0.390 − 0.390i)7-s + 0.378i·11-s + (−3.06 + 3.06i)13-s + (5.04 − 5.04i)17-s − 1.94i·19-s + (−0.707 − 0.707i)23-s + (1.45 − 4.78i)25-s − 0.264·29-s + 0.305·31-s + (1.22 + 0.182i)35-s + (1.13 + 1.13i)37-s + 4.48i·41-s + (−2.13 + 2.13i)43-s + (5.21 − 5.21i)47-s + ⋯
L(s)  = 1  + (−0.803 + 0.595i)5-s + (−0.147 − 0.147i)7-s + 0.114i·11-s + (−0.848 + 0.848i)13-s + (1.22 − 1.22i)17-s − 0.447i·19-s + (−0.147 − 0.147i)23-s + (0.291 − 0.956i)25-s − 0.0491·29-s + 0.0548·31-s + (0.206 + 0.0307i)35-s + (0.187 + 0.187i)37-s + 0.700i·41-s + (−0.325 + 0.325i)43-s + (0.761 − 0.761i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.522 - 0.852i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.261093027\)
\(L(\frac12)\) \(\approx\) \(1.261093027\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 - 1.33i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (0.390 + 0.390i)T + 7iT^{2} \)
11 \( 1 - 0.378iT - 11T^{2} \)
13 \( 1 + (3.06 - 3.06i)T - 13iT^{2} \)
17 \( 1 + (-5.04 + 5.04i)T - 17iT^{2} \)
19 \( 1 + 1.94iT - 19T^{2} \)
29 \( 1 + 0.264T + 29T^{2} \)
31 \( 1 - 0.305T + 31T^{2} \)
37 \( 1 + (-1.13 - 1.13i)T + 37iT^{2} \)
41 \( 1 - 4.48iT - 41T^{2} \)
43 \( 1 + (2.13 - 2.13i)T - 43iT^{2} \)
47 \( 1 + (-5.21 + 5.21i)T - 47iT^{2} \)
53 \( 1 + (-9.69 - 9.69i)T + 53iT^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + (1.52 + 1.52i)T + 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + (6.79 - 6.79i)T - 73iT^{2} \)
79 \( 1 + 3.47iT - 79T^{2} \)
83 \( 1 + (5.49 + 5.49i)T + 83iT^{2} \)
89 \( 1 - 9.03T + 89T^{2} \)
97 \( 1 + (-0.595 - 0.595i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.485685992456050363609839712687, −7.55615582193701603825370170018, −7.21488160653564471587758063479, −6.56950377280867725974703857891, −5.51177286665786341162834322295, −4.69800348252045193800422187456, −3.97519387870975053500252329560, −3.05240492065385069458847996656, −2.35306934480135335157885136738, −0.820702517071723870793153071009, 0.50066037140714451721306177754, 1.67146668663098332902751581373, 2.95926287282431679502083410851, 3.70451924088705379578779552941, 4.45439714977539397050222329743, 5.47041842242541578819776674764, 5.81974068028505288457016989573, 7.03043127070465534176002074954, 7.72873555461067549181075663879, 8.178932444086718854763645964489

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