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Properties

Label 2-2070-15.8-c1-0-34
Degree $2$
Conductor $2070$
Sign $0.956 + 0.292i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.956 + 0.292i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.074215410\)
\(L(\frac12)\) \(\approx\) \(3.074215410\)
\(L(\frac{3}{2})\) 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

−8.803197896807787354859383144374, −8.226698550515885235486267257445, −7.63120569891999366327413125460, −6.63718459689082633499236756250, −5.70565301914907346528677425887, −5.26261215892902272625033004041, −4.09333723122210156119357264922, −3.74729203512368312210715675161, −1.96614876265035131254256036623, −1.02585947634109265779837056491, 1.48541004813673635188895058886, 2.37114639707914038908708293635, 3.01550928782990874729674174999, 4.37079361472088026425309906622, 5.24335290741421057227873766628, 5.67784246600987815733096515808, 6.73145268710343502979963428098, 7.50654114227499572669942403725, 8.510198961500849115193805284122, 9.313267424865284763330711612825

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