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Properties

Label 2-2070-15.2-c1-0-15
Degree $2$
Conductor $2070$
Sign $0.162 - 0.986i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.226 + 2.22i)5-s + (−2.57 − 2.57i)7-s + (0.707 + 0.707i)8-s + (−1.41 − 1.73i)10-s + 4.44i·11-s + (2 − 2i)13-s + 3.64·14-s − 1.00·16-s + (4.09 − 4.09i)17-s + 0.825i·19-s + (2.22 + 0.226i)20-s + (−3.14 − 3.14i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.101 + 0.994i)5-s + (−0.974 − 0.974i)7-s + (0.250 + 0.250i)8-s + (−0.446 − 0.548i)10-s + 1.34i·11-s + (0.554 − 0.554i)13-s + 0.974·14-s − 0.250·16-s + (0.993 − 0.993i)17-s + 0.189i·19-s + (0.497 + 0.0506i)20-s + (−0.670 − 0.670i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.077642046\)
\(L(\frac12)\) \(\approx\) \(1.077642046\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (0.226 - 2.22i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (2.57 + 2.57i)T + 7iT^{2} \)
11 \( 1 - 4.44iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (-4.09 + 4.09i)T - 17iT^{2} \)
19 \( 1 - 0.825iT - 19T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 - 5.79T + 31T^{2} \)
37 \( 1 + (-1.43 - 1.43i)T + 37iT^{2} \)
41 \( 1 + 3.57iT - 41T^{2} \)
43 \( 1 + (0.569 - 0.569i)T - 43iT^{2} \)
47 \( 1 + (1.86 - 1.86i)T - 47iT^{2} \)
53 \( 1 + (-2.80 - 2.80i)T + 53iT^{2} \)
59 \( 1 + 5.30T + 59T^{2} \)
61 \( 1 - 0.328T + 61T^{2} \)
67 \( 1 + (-9.53 - 9.53i)T + 67iT^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + (3.90 - 3.90i)T - 73iT^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + (-9.36 - 9.36i)T + 83iT^{2} \)
89 \( 1 - 8.63T + 89T^{2} \)
97 \( 1 + (6.39 + 6.39i)T + 97iT^{2} \)
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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

−9.646360724201585045980604057077, −8.323274670834836599340590395155, −7.53061707584293520897318812796, −7.03445627573684406032432227207, −6.46619583149818485064593063297, −5.53588846742591423607184204183, −4.37480181423009923856802052692, −3.45542761378337020091636587127, −2.52248317236254648622619538468, −0.922482461974039400138209317546, 0.60977528581206708844911088781, 1.79263777518447788330454006734, 3.10181611179591373094270997411, 3.70535226212405363354896195322, 4.89814487234770049725931184803, 6.00797420950582281704308784847, 6.27405375707011481854735797926, 7.78431181335495462718472527900, 8.424976610994475781214241590098, 8.946092737793428390501062474025

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