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Properties

Label 2-2475-5.4-c1-0-41
Degree $2$
Conductor $2475$
Sign $0.447 - 0.894i$
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  + 2i·2-s − 2·4-s − 2i·7-s − 11-s − 4i·13-s + 4·14-s − 4·16-s + 2i·17-s − 2i·22-s − i·23-s + 8·26-s + 4i·28-s + 7·31-s − 8i·32-s − 4·34-s + ⋯
L(s)  = 1  + 1.41i·2-s − 4-s − 0.755i·7-s − 0.301·11-s − 1.10i·13-s + 1.06·14-s − 16-s + 0.485i·17-s − 0.426i·22-s − 0.208i·23-s + 1.56·26-s + 0.755i·28-s + 1.25·31-s − 1.41i·32-s − 0.685·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
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: \((2,\ 2475,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.638542166\)
\(L(\frac12)\) \(\approx\) \(1.638542166\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 2T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 5T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−8.660696378935712273189641518391, −8.067347954202489738830910533190, −7.58060360641750635872273022794, −6.74198575776120250887442349566, −6.12413043052292813165365685578, −5.29958517375224659287044402422, −4.59718540584982485388701259771, −3.58582445546653028719375067578, −2.39339510032273414591697618803, −0.69904508467170606839526776832, 0.955373789394485655072388047414, 2.20193489479546506633400668487, 2.67544941524919290821879443709, 3.80720158844220703298532645371, 4.53131206918408403275588185363, 5.48839480308999129741362162432, 6.46507675682819691012914834694, 7.26801250680856411924011612348, 8.328148512145031190969761258208, 9.165084479317677098385494198288

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