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Properties

Label 2-2070-15.2-c1-0-34
Degree $2$
Conductor $2070$
Sign $0.0358 + 0.999i$
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 + (2.02 − 0.947i)5-s + (0.707 + 0.707i)8-s + (−0.762 + 2.10i)10-s − 4.20i·11-s + (−1.52 + 1.52i)13-s − 1.00·16-s + (−3.30 + 3.30i)17-s − 7.93i·19-s + (−0.947 − 2.02i)20-s + (2.97 + 2.97i)22-s + (0.707 + 0.707i)23-s + (3.20 − 3.83i)25-s − 2.15i·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.905 − 0.423i)5-s + (0.250 + 0.250i)8-s + (−0.241 + 0.664i)10-s − 1.26i·11-s + (−0.422 + 0.422i)13-s − 0.250·16-s + (−0.802 + 0.802i)17-s − 1.81i·19-s + (−0.211 − 0.452i)20-s + (0.634 + 0.634i)22-s + (0.147 + 0.147i)23-s + (0.640 − 0.767i)25-s − 0.422i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0358 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0358 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.0358 + 0.999i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.0358 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082945066\)
\(L(\frac12)\) \(\approx\) \(1.082945066\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 0.947i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 + 4.20iT - 11T^{2} \)
13 \( 1 + (1.52 - 1.52i)T - 13iT^{2} \)
17 \( 1 + (3.30 - 3.30i)T - 17iT^{2} \)
19 \( 1 + 7.93iT - 19T^{2} \)
29 \( 1 - 2.98T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (5.06 + 5.06i)T + 37iT^{2} \)
41 \( 1 + 1.93iT - 41T^{2} \)
43 \( 1 + (-1.70 + 1.70i)T - 43iT^{2} \)
47 \( 1 + (0.155 - 0.155i)T - 47iT^{2} \)
53 \( 1 + (-5.80 - 5.80i)T + 53iT^{2} \)
59 \( 1 - 3.27T + 59T^{2} \)
61 \( 1 + 9.20T + 61T^{2} \)
67 \( 1 + (6.95 + 6.95i)T + 67iT^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (2.41 - 2.41i)T - 73iT^{2} \)
79 \( 1 - 1.72iT - 79T^{2} \)
83 \( 1 + (7.65 + 7.65i)T + 83iT^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (-8.57 - 8.57i)T + 97iT^{2} \)
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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

−8.928202540448874902759837046629, −8.456614620505752667405955363842, −7.24697728920618880960181529146, −6.62810628811134956327906688901, −5.80736400444116779433908895343, −5.16394288844927363713452425697, −4.19353827276720062149689418003, −2.79383751739857558156146751554, −1.77389620974939843765403308395, −0.43578192633168323538430700244, 1.51458433619607404008339076951, 2.30053190797362580057995647541, 3.22534127795717679425021861873, 4.41338641760631085243625741815, 5.28442354706930773429760025369, 6.27222275616201114977682963878, 7.11351577482180793931499215241, 7.70320468662037276509415874365, 8.755390529303609045078433610086, 9.468394733338194722857183784959

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