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Properties

Label 2-2070-15.8-c1-0-13
Degree $2$
Conductor $2070$
Sign $0.594 - 0.803i$
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 + (−1.43 + 1.71i)5-s + (3.17 − 3.17i)7-s + (0.707 − 0.707i)8-s + (2.22 − 0.193i)10-s + 3.42i·11-s + (2 + 2i)13-s − 4.48·14-s − 1.00·16-s + (−1.61 − 1.61i)17-s + 6.45i·19-s + (−1.71 − 1.43i)20-s + (2.42 − 2.42i)22-s + (−0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.643 + 0.765i)5-s + (1.19 − 1.19i)7-s + (0.250 − 0.250i)8-s + (0.704 − 0.0611i)10-s + 1.03i·11-s + (0.554 + 0.554i)13-s − 1.19·14-s − 0.250·16-s + (−0.390 − 0.390i)17-s + 1.48i·19-s + (−0.382 − 0.321i)20-s + (0.516 − 0.516i)22-s + (−0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.594 - 0.803i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.594 - 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.161148873\)
\(L(\frac12)\) \(\approx\) \(1.161148873\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 - 1.71i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-3.17 + 3.17i)T - 7iT^{2} \)
11 \( 1 - 3.42iT - 11T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (1.61 + 1.61i)T + 17iT^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 + 2.27T + 31T^{2} \)
37 \( 1 + (-1.24 + 1.24i)T - 37iT^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 + (0.752 + 0.752i)T + 43iT^{2} \)
47 \( 1 + (4.29 + 4.29i)T + 47iT^{2} \)
53 \( 1 + (2.08 - 2.08i)T - 53iT^{2} \)
59 \( 1 + 7.47T + 59T^{2} \)
61 \( 1 + 3.89T + 61T^{2} \)
67 \( 1 + (-8.93 + 8.93i)T - 67iT^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-6.06 - 6.06i)T + 73iT^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 + (-9.32 + 9.32i)T - 83iT^{2} \)
89 \( 1 - 8.03T + 89T^{2} \)
97 \( 1 + (11.3 - 11.3i)T - 97iT^{2} \)
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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

−9.388206906573864924166057020487, −8.109597880275559220520375565859, −7.934891935527172466925711725704, −7.08619935171147930569797446429, −6.43422594650693591934999026461, −4.86870552286581790005879173537, −4.18164794500496369787134015888, −3.51996847400407132238291961208, −2.15491975622682706589725017861, −1.23355146312957017788571298370, 0.54720324307167838349774826148, 1.77627717231236159592153875847, 3.04343004873129482653352040397, 4.37266769139562357297823082257, 5.15017001408028799706520294352, 5.72860157779250331591037720433, 6.66509888264544028292245620707, 7.85750371546075021575155098259, 8.243221860198899819717052208211, 8.876116187637669599791442290811

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