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Properties

Label 2-2070-15.8-c1-0-16
Degree $2$
Conductor $2070$
Sign $0.749 - 0.662i$
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 + (0.707 + 2.12i)5-s + (0.707 − 0.707i)8-s + (0.999 − 2i)10-s − 4.24i·11-s + (2 + 2i)13-s − 1.00·16-s + (4.24 + 4.24i)17-s + 4i·19-s + (−2.12 + 0.707i)20-s + (−3 + 3i)22-s + (0.707 − 0.707i)23-s + (−3.99 + 3i)25-s − 2.82i·26-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.316 + 0.948i)5-s + (0.250 − 0.250i)8-s + (0.316 − 0.632i)10-s − 1.27i·11-s + (0.554 + 0.554i)13-s − 0.250·16-s + (1.02 + 1.02i)17-s + 0.917i·19-s + (−0.474 + 0.158i)20-s + (−0.639 + 0.639i)22-s + (0.147 − 0.147i)23-s + (−0.799 + 0.600i)25-s − 0.554i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.749 - 0.662i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.410580935\)
\(L(\frac12)\) \(\approx\) \(1.410580935\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 2.12i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 + (-5.65 + 5.65i)T - 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (-11 - 11i)T + 73iT^{2} \)
79 \( 1 - 16iT - 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 + (12 - 12i)T - 97iT^{2} \)
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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

−9.283909523190332207630939265667, −8.361670460814043542556185815524, −7.955815995981993917005111374426, −6.80205799846479615003767511696, −6.18038717125954618649546589111, −5.40090650636619287780921819912, −3.78181627549545089838608103180, −3.44149995007849676074060164821, −2.28095801537522642687306904248, −1.18013406386781756225502901628, 0.68674721752069848472401139000, 1.76656026089054311399794108084, 3.04843542330823899172813465592, 4.51672065068427047269720500087, 5.02501154495949726443363274857, 5.85987756796459168767111323446, 6.78428728747704863068806071862, 7.59857629553015041620868435874, 8.210276192927650985860874605787, 9.090246873746707589569277780860

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