↑

Properties

Label 2-4140-15.8-c1-0-16
Degree $2$
Conductor $4140$
Sign $0.00877 - 0.999i$
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  + (0.818 + 2.08i)5-s + (0.623 − 0.623i)7-s + 1.13i·11-s + (1.31 + 1.31i)13-s + (−4.45 − 4.45i)17-s + 0.624i·19-s + (−0.707 + 0.707i)23-s + (−3.65 + 3.40i)25-s + 3.20·29-s + 9.14·31-s + (1.80 + 0.787i)35-s + (−0.268 + 0.268i)37-s + 1.26i·41-s + (7.06 + 7.06i)43-s + (1.93 + 1.93i)47-s + ⋯
L(s)  = 1  + (0.366 + 0.930i)5-s + (0.235 − 0.235i)7-s + 0.341i·11-s + (0.364 + 0.364i)13-s + (−1.08 − 1.08i)17-s + 0.143i·19-s + (−0.147 + 0.147i)23-s + (−0.731 + 0.681i)25-s + 0.595·29-s + 1.64·31-s + (0.305 + 0.133i)35-s + (−0.0441 + 0.0441i)37-s + 0.197i·41-s + (1.07 + 1.07i)43-s + (0.282 + 0.282i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00877 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00877 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.00877 - 0.999i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.00877 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.829091539\)
\(L(\frac12)\) \(\approx\) \(1.829091539\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.818 - 2.08i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-0.623 + 0.623i)T - 7iT^{2} \)
11 \( 1 - 1.13iT - 11T^{2} \)
13 \( 1 + (-1.31 - 1.31i)T + 13iT^{2} \)
17 \( 1 + (4.45 + 4.45i)T + 17iT^{2} \)
19 \( 1 - 0.624iT - 19T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 - 9.14T + 31T^{2} \)
37 \( 1 + (0.268 - 0.268i)T - 37iT^{2} \)
41 \( 1 - 1.26iT - 41T^{2} \)
43 \( 1 + (-7.06 - 7.06i)T + 43iT^{2} \)
47 \( 1 + (-1.93 - 1.93i)T + 47iT^{2} \)
53 \( 1 + (1.62 - 1.62i)T - 53iT^{2} \)
59 \( 1 + 4.50T + 59T^{2} \)
61 \( 1 - 0.362T + 61T^{2} \)
67 \( 1 + (4.96 - 4.96i)T - 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-7.41 - 7.41i)T + 73iT^{2} \)
79 \( 1 - 9.11iT - 79T^{2} \)
83 \( 1 + (6.52 - 6.52i)T - 83iT^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + (-1.67 + 1.67i)T - 97iT^{2} \)
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

−8.568334434543514079186825703500, −7.77934528663965570150732842373, −7.03708907314668404490065228389, −6.50424408207834472829030285655, −5.80609703843340279027212003526, −4.70658609723782943272668873512, −4.16343160325461367807761787457, −2.94810967825106207156958912073, −2.39651641325897808773935677007, −1.18475877240634149914064584635, 0.55138481234884554342858282709, 1.68021507559857885141717332579, 2.55299468405457497016543389239, 3.75719560779999038335872252413, 4.52673585700859685021603185636, 5.21887723946893398364007505839, 6.08899954298005757485991676647, 6.51574238431435077843778547113, 7.73372464029006665572775312005, 8.400815283671215647971522683640

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