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Properties

Label 2-4140-15.2-c1-0-16
Degree $2$
Conductor $4140$
Sign $0.978 + 0.204i$
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  + (−1.79 + 1.33i)5-s + (−2.07 − 2.07i)7-s + 3.22i·11-s + (4.92 − 4.92i)13-s + (−3.44 + 3.44i)17-s + 1.09i·19-s + (0.707 + 0.707i)23-s + (1.41 − 4.79i)25-s − 6.42·29-s − 5.08·31-s + (6.49 + 0.941i)35-s + (3.18 + 3.18i)37-s + 3.93i·41-s + (−0.659 + 0.659i)43-s + (9.43 − 9.43i)47-s + ⋯
L(s)  = 1  + (−0.801 + 0.598i)5-s + (−0.784 − 0.784i)7-s + 0.972i·11-s + (1.36 − 1.36i)13-s + (−0.834 + 0.834i)17-s + 0.250i·19-s + (0.147 + 0.147i)23-s + (0.283 − 0.958i)25-s − 1.19·29-s − 0.912·31-s + (1.09 + 0.159i)35-s + (0.523 + 0.523i)37-s + 0.613i·41-s + (−0.100 + 0.100i)43-s + (1.37 − 1.37i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.978 + 0.204i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.978 + 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.187919737\)
\(L(\frac12)\) \(\approx\) \(1.187919737\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 2.07i)T + 7iT^{2} \)
11 \( 1 - 3.22iT - 11T^{2} \)
13 \( 1 + (-4.92 + 4.92i)T - 13iT^{2} \)
17 \( 1 + (3.44 - 3.44i)T - 17iT^{2} \)
19 \( 1 - 1.09iT - 19T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + (-3.18 - 3.18i)T + 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (0.659 - 0.659i)T - 43iT^{2} \)
47 \( 1 + (-9.43 + 9.43i)T - 47iT^{2} \)
53 \( 1 + (5.54 + 5.54i)T + 53iT^{2} \)
59 \( 1 - 2.11T + 59T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + (-7.76 - 7.76i)T + 67iT^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 + (7.21 - 7.21i)T - 73iT^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 + (-0.143 - 0.143i)T + 83iT^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + (-5.05 - 5.05i)T + 97iT^{2} \)
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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

−8.258138694728817305438589242216, −7.61188730918457633337266214717, −6.98210742458099914022374030302, −6.33209823865248683875009481387, −5.54786051819409642811470872133, −4.35121261395485595444564640473, −3.70474601068756475365529665960, −3.22375133703554279627984089201, −1.93647958498854984090032895601, −0.55080236833522027947489332429, 0.67454643765839903208142944790, 1.99263624709008268255247917785, 3.12645121332877158740413704479, 3.84673341505126826124075489936, 4.55344289715650892561001525785, 5.62128155489152343214728544373, 6.16237318341254038504506858917, 6.96819629354659633059161188073, 7.73925818350950655671423996349, 8.718927482186068522721689992805

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