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Properties

Label 2-4140-15.8-c1-0-10
Degree $2$
Conductor $4140$
Sign $0.511 - 0.859i$
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  + (0.340 − 2.21i)5-s + (−3.49 + 3.49i)7-s − 1.20i·11-s + (0.0289 + 0.0289i)13-s + (4.88 + 4.88i)17-s − 6.86i·19-s + (0.707 − 0.707i)23-s + (−4.76 − 1.50i)25-s − 2.41·29-s + 4.34·31-s + (6.53 + 8.91i)35-s + (−7.27 + 7.27i)37-s + 1.65i·41-s + (−2.38 − 2.38i)43-s + (−1.54 − 1.54i)47-s + ⋯
L(s)  = 1  + (0.152 − 0.988i)5-s + (−1.32 + 1.32i)7-s − 0.363i·11-s + (0.00802 + 0.00802i)13-s + (1.18 + 1.18i)17-s − 1.57i·19-s + (0.147 − 0.147i)23-s + (−0.953 − 0.300i)25-s − 0.449·29-s + 0.780·31-s + (1.10 + 1.50i)35-s + (−1.19 + 1.19i)37-s + 0.258i·41-s + (−0.362 − 0.362i)43-s + (−0.226 − 0.226i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.511 - 0.859i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.231674743\)
\(L(\frac12)\) \(\approx\) \(1.231674743\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.340 + 2.21i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (3.49 - 3.49i)T - 7iT^{2} \)
11 \( 1 + 1.20iT - 11T^{2} \)
13 \( 1 + (-0.0289 - 0.0289i)T + 13iT^{2} \)
17 \( 1 + (-4.88 - 4.88i)T + 17iT^{2} \)
19 \( 1 + 6.86iT - 19T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
31 \( 1 - 4.34T + 31T^{2} \)
37 \( 1 + (7.27 - 7.27i)T - 37iT^{2} \)
41 \( 1 - 1.65iT - 41T^{2} \)
43 \( 1 + (2.38 + 2.38i)T + 43iT^{2} \)
47 \( 1 + (1.54 + 1.54i)T + 47iT^{2} \)
53 \( 1 + (-0.188 + 0.188i)T - 53iT^{2} \)
59 \( 1 - 0.182T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + (4.11 - 4.11i)T - 67iT^{2} \)
71 \( 1 - 8.99iT - 71T^{2} \)
73 \( 1 + (-5.12 - 5.12i)T + 73iT^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (7.95 - 7.95i)T - 83iT^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + (-3.45 + 3.45i)T - 97iT^{2} \)
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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

−8.612412925138177567956933190627, −8.121075181954555347898393589403, −6.86433295805456651864454028822, −6.28640828145436933235573481201, −5.50505309481326654427280746101, −5.06977931497249170157695133839, −3.85650049156827672359256537123, −3.07930153404024590828079191940, −2.20465672493565387897974280272, −0.929509543507700545951198730053, 0.42554274131814503852644786690, 1.80048419705120287921673595704, 3.17518748218903022600031513914, 3.40439273188286953123312851491, 4.31512626874748654263038847411, 5.51642888747927919410799007151, 6.19814964640222858099031371579, 6.98046516232165042140170992397, 7.37510785144790714154099496027, 8.008997873641722066908036637826

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