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Properties

Label 2-3060-1020.539-c0-0-1
Degree $2$
Conductor $3060$
Sign $0.694 - 0.719i$
Analytic cond. $1.52713$
Root an. cond. $1.23577$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s − 5-s + (−0.923 − 0.382i)8-s + (−0.382 − 0.923i)10-s + (1.30 − 1.30i)13-s − i·16-s − i·17-s + (0.707 − 0.707i)20-s + 25-s + (1.70 + 0.707i)26-s + (1.08 + 0.216i)29-s + (0.923 − 0.382i)32-s + (0.923 − 0.382i)34-s + (1.08 + 1.63i)37-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s − 5-s + (−0.923 − 0.382i)8-s + (−0.382 − 0.923i)10-s + (1.30 − 1.30i)13-s − i·16-s − i·17-s + (0.707 − 0.707i)20-s + 25-s + (1.70 + 0.707i)26-s + (1.08 + 0.216i)29-s + (0.923 − 0.382i)32-s + (0.923 − 0.382i)34-s + (1.08 + 1.63i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(1.52713\)
Root analytic conductor: \(1.23577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :0),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168616411\)
\(L(\frac12)\) \(\approx\) \(1.168616411\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
3 \( 1 \)
5 \( 1 + T \)
17 \( 1 + iT \)
good7 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
97 \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \)
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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

−8.617349134362157289865675365528, −8.035328961504202028158045479545, −7.69165087996865487048207222738, −6.57966531639629261471930604021, −6.17687071799353634632043838104, −4.98237789173602987869779586087, −4.59870848283750236770241111094, −3.35165022392794437876663815213, −3.08422171470270937075315832408, −0.837164186519551019568903931852, 1.08609718053162245066892200009, 2.20127463200836206351333429155, 3.35665107049196180203108917192, 4.09951404734899739134137478202, 4.44954819806591707125104958441, 5.72305260447145010020301784693, 6.36056340288467848151718159879, 7.30940152841099940603653157950, 8.366115008663031338377113637329, 8.776633408868741357684798380966

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