Properties

Label 2-2475-495.439-c0-0-0
Degree $2$
Conductor $2475$
Sign $-0.803 - 0.595i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.499i)12-s + (−0.499 + 0.866i)16-s + (−1.73 + i)23-s + 0.999i·27-s + (0.5 + 0.866i)31-s − 0.999i·33-s + 0.999·36-s i·37-s − 0.999·44-s + (0.866 + 0.5i)47-s − 0.999i·48-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.499i)12-s + (−0.499 + 0.866i)16-s + (−1.73 + i)23-s + 0.999i·27-s + (0.5 + 0.866i)31-s − 0.999i·33-s + 0.999·36-s i·37-s − 0.999·44-s + (0.866 + 0.5i)47-s − 0.999i·48-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1924, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7499758754\)
\(L(\frac12)\) \(\approx\) \(0.7499758754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.528327767847837080065683623844, −8.671557563247609180192398313553, −7.62895125750952989987674137709, −7.26415791938763102600590165139, −6.25186879635549421369336562786, −5.62088639152425492222226990586, −4.53069524800404915178777638018, −3.95710386230969074382491729595, −2.90835925064399848199713132426, −1.74284452052943568260306113996, 0.53115092429659283065575484937, 1.81241593553954901483905139104, 2.71049156770700380527076515308, 4.19964707562303371141548440401, 5.10602467989208173142266921819, 5.93228986506209758714463628409, 6.23226947785887894819255443348, 7.13396606402587707066788029906, 7.934409906643424644846018266745, 8.689533112582305679743833974213

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