Properties

Label 2-2070-23.22-c2-0-15
Degree $2$
Conductor $2070$
Sign $-0.196 - 0.980i$
Analytic cond. $56.4034$
Root an. cond. $7.51022$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s − 2.23i·5-s + 7.61i·7-s + 2.82·8-s − 3.16i·10-s − 12.3i·11-s − 13.0·13-s + 10.7i·14-s + 4.00·16-s + 9.13i·17-s + 14.4i·19-s − 4.47i·20-s − 17.5i·22-s + (−22.5 + 4.51i)23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s − 0.447i·5-s + 1.08i·7-s + 0.353·8-s − 0.316i·10-s − 1.12i·11-s − 1.00·13-s + 0.769i·14-s + 0.250·16-s + 0.537i·17-s + 0.760i·19-s − 0.223i·20-s − 0.795i·22-s + (−0.980 + 0.196i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(56.4034\)
Root analytic conductor: \(7.51022\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1),\ -0.196 - 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.056160308\)
\(L(\frac12)\) \(\approx\) \(2.056160308\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + 2.23iT \)
23 \( 1 + (22.5 - 4.51i)T \)
good7 \( 1 - 7.61iT - 49T^{2} \)
11 \( 1 + 12.3iT - 121T^{2} \)
13 \( 1 + 13.0T + 169T^{2} \)
17 \( 1 - 9.13iT - 289T^{2} \)
19 \( 1 - 14.4iT - 361T^{2} \)
29 \( 1 - 21.2T + 841T^{2} \)
31 \( 1 - 36.8T + 961T^{2} \)
37 \( 1 - 56.9iT - 1.36e3T^{2} \)
41 \( 1 + 70.7T + 1.68e3T^{2} \)
43 \( 1 - 70.0iT - 1.84e3T^{2} \)
47 \( 1 - 66.2T + 2.20e3T^{2} \)
53 \( 1 - 77.4iT - 2.80e3T^{2} \)
59 \( 1 + 82.7T + 3.48e3T^{2} \)
61 \( 1 - 23.9iT - 3.72e3T^{2} \)
67 \( 1 + 118. iT - 4.48e3T^{2} \)
71 \( 1 + 69.0T + 5.04e3T^{2} \)
73 \( 1 - 25.9T + 5.32e3T^{2} \)
79 \( 1 + 28.8iT - 6.24e3T^{2} \)
83 \( 1 - 69.3iT - 6.88e3T^{2} \)
89 \( 1 - 45.4iT - 7.92e3T^{2} \)
97 \( 1 + 74.4iT - 9.40e3T^{2} \)
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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

−9.067662165142380162723801025015, −8.263746919589569598823148659566, −7.79167295432903941594036423220, −6.37339228275920593078785380609, −6.02314889068851007264653116775, −5.14183045874978412443471144582, −4.42304577744159827863688905788, −3.27730550648666467848518208167, −2.51698363020602713038786430970, −1.34320420826123935035279364696, 0.37857188217833914011164415925, 1.97051172887879008223372212542, 2.79670329558371116266606271451, 3.95733156837530657884427188072, 4.55760870605969163489743475605, 5.34142777860359877171367957008, 6.52432439316862094866257840976, 7.19633355272475210541013273841, 7.47084181652063181788695947729, 8.658193820328130688488240547974

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