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Properties

Label 2-2070-15.2-c1-0-2
Degree $2$
Conductor $2070$
Sign $0.239 - 0.970i$
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 + (0.401 − 2.19i)5-s + (−0.707 − 0.707i)8-s + (−1.27 − 1.83i)10-s + 6.12i·11-s + (−2.54 + 2.54i)13-s − 1.00·16-s + (−2.98 + 2.98i)17-s + 6.81i·19-s + (−2.19 − 0.401i)20-s + (4.33 + 4.33i)22-s + (−0.707 − 0.707i)23-s + (−4.67 − 1.76i)25-s + 3.59i·26-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.179 − 0.983i)5-s + (−0.250 − 0.250i)8-s + (−0.402 − 0.581i)10-s + 1.84i·11-s + (−0.705 + 0.705i)13-s − 0.250·16-s + (−0.724 + 0.724i)17-s + 1.56i·19-s + (−0.491 − 0.0896i)20-s + (0.923 + 0.923i)22-s + (−0.147 − 0.147i)23-s + (−0.935 − 0.352i)25-s + 0.705i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.239 - 0.970i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.239 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.037245159\)
\(L(\frac12)\) \(\approx\) \(1.037245159\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.401 + 2.19i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 6.12iT - 11T^{2} \)
13 \( 1 + (2.54 - 2.54i)T - 13iT^{2} \)
17 \( 1 + (2.98 - 2.98i)T - 17iT^{2} \)
19 \( 1 - 6.81iT - 19T^{2} \)
29 \( 1 + 9.75T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (-6.24 - 6.24i)T + 37iT^{2} \)
41 \( 1 + 6.28iT - 41T^{2} \)
43 \( 1 + (8.55 - 8.55i)T - 43iT^{2} \)
47 \( 1 + (-6.92 + 6.92i)T - 47iT^{2} \)
53 \( 1 + (0.227 + 0.227i)T + 53iT^{2} \)
59 \( 1 - 1.10T + 59T^{2} \)
61 \( 1 + 6.07T + 61T^{2} \)
67 \( 1 + (-9.14 - 9.14i)T + 67iT^{2} \)
71 \( 1 + 6.30iT - 71T^{2} \)
73 \( 1 + (-1.35 + 1.35i)T - 73iT^{2} \)
79 \( 1 + 5.13iT - 79T^{2} \)
83 \( 1 + (2.28 + 2.28i)T + 83iT^{2} \)
89 \( 1 - 9.75T + 89T^{2} \)
97 \( 1 + (5.12 + 5.12i)T + 97iT^{2} \)
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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

−9.614925062074586730957643266353, −8.645150502040175414597507796865, −7.70066808225731704086253120169, −6.91860081669275219627612690844, −5.90915363971238403997356457337, −5.09914039459093287951996044123, −4.36280739359883138734793247472, −3.78209257978813223892153050723, −1.97990202404138669635497622415, −1.80940646096327592471648440972, 0.27892894177034214238214480799, 2.41808811978086228029288154179, 3.08535891436413608658861535408, 3.93674281399653831829870051737, 5.18769164974873645781668258511, 5.76423154890778990473339580651, 6.57180226860471151366186003950, 7.32787297381736554076591669441, 7.905695959455156329680683028278, 9.026691558318562744593286513059

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