Properties

Label 2-2475-5.4-c1-0-63
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  + i·2-s + 4-s − 4i·7-s + 3i·8-s − 11-s − 2i·13-s + 4·14-s − 16-s − 2i·17-s i·22-s − 8i·23-s + 2·26-s − 4i·28-s − 6·29-s − 8·31-s + 5i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.5·4-s − 1.51i·7-s + 1.06i·8-s − 0.301·11-s − 0.554i·13-s + 1.06·14-s − 0.250·16-s − 0.485i·17-s − 0.213i·22-s − 1.66i·23-s + 0.392·26-s − 0.755i·28-s − 1.11·29-s − 1.43·31-s + 0.883i·32-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

\(L(1)\) \(\approx\) \(1.543697105\)
\(L(\frac12)\) \(\approx\) \(1.543697105\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 2T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.572118508735431150178297917237, −7.65425551010714806031984579471, −7.39237206538559556034207148451, −6.62739717935609349818089283314, −5.81035351536980736913727158654, −4.96266235526207981575931562842, −4.05335386818013800255305889111, −3.05111798410349557260718715854, −1.93158060445720587440926283626, −0.46773758599670833236951919248, 1.59218125761055456862938865920, 2.22233493526386804438013593533, 3.18560476081113685080029332595, 3.96787474701537084908378451881, 5.35436801606081535945364348358, 5.79021730882807693922797072594, 6.76075008149238254837919490492, 7.52416623741558427761001439276, 8.426414204575432439515965727940, 9.313090476492710248391318116837

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