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Properties

Label 2-2070-15.8-c1-0-10
Degree $2$
Conductor $2070$
Sign $0.927 - 0.374i$
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 + (−2.12 + 0.707i)5-s + (0.707 − 0.707i)8-s + (2 + 0.999i)10-s + 1.41i·11-s + (−4 − 4i)13-s − 1.00·16-s + (4.24 + 4.24i)17-s − 6i·19-s + (−0.707 − 2.12i)20-s + (1.00 − 1.00i)22-s + (−0.707 + 0.707i)23-s + (3.99 − 3i)25-s + 5.65i·26-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.948 + 0.316i)5-s + (0.250 − 0.250i)8-s + (0.632 + 0.316i)10-s + 0.426i·11-s + (−1.10 − 1.10i)13-s − 0.250·16-s + (1.02 + 1.02i)17-s − 1.37i·19-s + (−0.158 − 0.474i)20-s + (0.213 − 0.213i)22-s + (−0.147 + 0.147i)23-s + (0.799 − 0.600i)25-s + 1.10i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.927 - 0.374i$
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: \((2,\ 2070,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8235842014\)
\(L(\frac12)\) \(\approx\) \(0.8235842014\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.707i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (4 + 4i)T + 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + (-1 - i)T + 43iT^{2} \)
47 \( 1 + (-7.07 - 7.07i)T + 47iT^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (9 - 9i)T - 67iT^{2} \)
71 \( 1 + 4.24iT - 71T^{2} \)
73 \( 1 + (-9 - 9i)T + 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (-8 + 8i)T - 97iT^{2} \)
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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

−9.250449181612321883334645599852, −8.329775454718213244695858831442, −7.53316165721037262579441569719, −7.33836967500827926832879988840, −6.05635695614761348083701770015, −4.99468914634203065979278023666, −4.10515945459695277538389417201, −3.18305871083576338010192924755, −2.39201440081612038950186814774, −0.826572020459474788265074205898, 0.49593222515311397664702242977, 1.92293683626551237046621158853, 3.32614721623000323471617765138, 4.25266578042254966735137015079, 5.14127037125086544741895854242, 5.91473664439364794082729233676, 7.09675740847605359648144276699, 7.46721684973520586020225502295, 8.215703504556490348158623597402, 9.018332037631596265485805297007

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