Properties

Label 2-2475-5.4-c1-0-72
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  − 1.73i·2-s − 0.999·4-s − 2i·7-s − 1.73i·8-s + 11-s + 5.46i·13-s − 3.46·14-s − 5·16-s − 5.46·19-s − 1.73i·22-s − 6.92i·23-s + 9.46·26-s + 1.99i·28-s − 3.46·29-s − 10.9·31-s + 5.19i·32-s + ⋯
L(s)  = 1  − 1.22i·2-s − 0.499·4-s − 0.755i·7-s − 0.612i·8-s + 0.301·11-s + 1.51i·13-s − 0.925·14-s − 1.25·16-s − 1.25·19-s − 0.369i·22-s − 1.44i·23-s + 1.85·26-s + 0.377i·28-s − 0.643·29-s − 1.96·31-s + 0.918i·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

\(L(1)\) \(\approx\) \(0.6860153426\)
\(L(\frac12)\) \(\approx\) \(0.6860153426\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT - 2T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 - 5.46iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.46T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 4.92iT - 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 4.92iT - 43T^{2} \)
47 \( 1 + 6.92iT - 47T^{2} \)
53 \( 1 + 0.928iT - 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 8.39iT - 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.953062455311298053551743563124, −7.58788386877263929289707635427, −6.78828699217701038954719185152, −6.32630323042012411979064576657, −4.83747016575679997910041884614, −4.09241325446444194011138237968, −3.56408390987197825658459308347, −2.24637538348886894622869036453, −1.64349476822250994938637824157, −0.20810773747728618761577760139, 1.79694761016219889223867217409, 2.88145826768297973328942596820, 3.97448122760600372419640997376, 5.18058364028652628663023411138, 5.69490244023560750193203742191, 6.21119806104241908709911456345, 7.33469465507360509724382511123, 7.67017042035867120358407241480, 8.624620470353080575662122675259, 9.045463865399637524196122466369

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