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Properties

Label 2-3060-1020.287-c0-0-1
Degree $2$
Conductor $3060$
Sign $0.137 - 0.990i$
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  − 2-s + 4-s + (−0.707 + 0.707i)5-s − 8-s + (0.707 − 0.707i)10-s + 16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)20-s − 1.00i·25-s + (0.707 − 0.292i)29-s − 32-s + (−0.707 − 0.707i)34-s + (−0.707 + 1.70i)37-s + (0.707 − 0.707i)40-s + (0.292 − 0.707i)41-s + ⋯
L(s)  = 1  − 2-s + 4-s + (−0.707 + 0.707i)5-s − 8-s + (0.707 − 0.707i)10-s + 16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)20-s − 1.00i·25-s + (0.707 − 0.292i)29-s − 32-s + (−0.707 − 0.707i)34-s + (−0.707 + 1.70i)37-s + (0.707 − 0.707i)40-s + (0.292 − 0.707i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.137 - 0.990i$
Analytic conductor: \(1.52713\)
Root analytic conductor: \(1.23577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :0),\ 0.137 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6104963332\)
\(L(\frac12)\) \(\approx\) \(0.6104963332\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.952963449811239030226959656218, −8.202084821308175164878814869022, −7.76928023786431637668167539559, −6.91140625681770110567899236093, −6.37289848549930400426223933687, −5.45831184823570248006052428398, −4.18828776709178147626351112332, −3.28510394514524721693767955953, −2.51774396371428459030250779395, −1.21640990348348868459427235847, 0.58787303426635366882585143981, 1.77195632754358503412352877187, 2.99537397880956825424067657657, 3.84227292391476725361307401179, 4.99381291142054941859876183487, 5.68653447243802228914580749557, 6.80983866529749061317646063580, 7.34089853070963044855837901159, 8.193309507465732105272925212198, 8.557636330814814290427668003250

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