Properties

Label 2-2475-5.4-c1-0-33
Degree $2$
Conductor $2475$
Sign $-0.894 + 0.447i$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
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  − 2.76i·2-s − 5.62·4-s − 1.86i·7-s + 10.0i·8-s − 11-s + 4.62i·13-s − 5.14·14-s + 16.4·16-s + 2.49i·17-s + 5.38·19-s + 2.76i·22-s − 7.14i·23-s + 12.7·26-s + 10.4i·28-s − 3.52·29-s + ⋯
L(s)  = 1  − 1.95i·2-s − 2.81·4-s − 0.704i·7-s + 3.54i·8-s − 0.301·11-s + 1.28i·13-s − 1.37·14-s + 4.10·16-s + 0.604i·17-s + 1.23·19-s + 0.588i·22-s − 1.49i·23-s + 2.50·26-s + 1.98i·28-s − 0.654·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.336580326\)
\(L(\frac12)\) \(\approx\) \(1.336580326\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 2.76iT - 2T^{2} \)
7 \( 1 + 1.86iT - 7T^{2} \)
13 \( 1 - 4.62iT - 13T^{2} \)
17 \( 1 - 2.49iT - 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 + 7.14iT - 23T^{2} \)
29 \( 1 + 3.52T + 29T^{2} \)
31 \( 1 - 8.62T + 31T^{2} \)
37 \( 1 - 8.87iT - 37T^{2} \)
41 \( 1 - 0.761T + 41T^{2} \)
43 \( 1 + 7.40iT - 43T^{2} \)
47 \( 1 + 0.373iT - 47T^{2} \)
53 \( 1 - 5.45iT - 53T^{2} \)
59 \( 1 - 5.14T + 59T^{2} \)
61 \( 1 - 4.42T + 61T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 - 6.77iT - 73T^{2} \)
79 \( 1 - 6.01T + 79T^{2} \)
83 \( 1 + 14.5iT - 83T^{2} \)
89 \( 1 - 9.04T + 89T^{2} \)
97 \( 1 + 16.3iT - 97T^{2} \)
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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

−8.848401824437264712022487223449, −8.255912817364448250404784452303, −7.24126010964467134052849166505, −6.09216054140344809363317340962, −4.89250653024869579753250941629, −4.37750956144432272704058928501, −3.57974394217545696398988694457, −2.67903730638321151792386517824, −1.72038707999507842265103927209, −0.70377265828950853634128280673, 0.818466543822623091284315871820, 2.92502133364387229471392122381, 3.88748490994798439627027479673, 5.11067138495709330918360779162, 5.43515068853693070366077472581, 6.05577297563978035319610800562, 7.08675836860646258605325479863, 7.69468948071611929163166483353, 8.151376952238876324373690723834, 9.085591297245701595363814208045

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