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Properties

Label 2-2475-2475.1517-c0-0-0
Degree $2$
Conductor $2475$
Sign $-0.879 - 0.475i$
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  + (−0.913 + 0.406i)3-s + (−0.994 + 0.104i)4-s + (−0.866 + 0.5i)5-s + (0.669 − 0.743i)9-s + (0.743 + 0.669i)11-s + (0.866 − 0.499i)12-s + (0.587 − 0.809i)15-s + (0.978 − 0.207i)16-s + (0.809 − 0.587i)20-s + (1.07 + 1.65i)23-s + (0.499 − 0.866i)25-s + (−0.309 + 0.951i)27-s + (−0.379 + 0.169i)31-s + (−0.951 − 0.309i)33-s + (−0.587 + 0.809i)36-s + (−1.72 + 0.877i)37-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)3-s + (−0.994 + 0.104i)4-s + (−0.866 + 0.5i)5-s + (0.669 − 0.743i)9-s + (0.743 + 0.669i)11-s + (0.866 − 0.499i)12-s + (0.587 − 0.809i)15-s + (0.978 − 0.207i)16-s + (0.809 − 0.587i)20-s + (1.07 + 1.65i)23-s + (0.499 − 0.866i)25-s + (−0.309 + 0.951i)27-s + (−0.379 + 0.169i)31-s + (−0.951 − 0.309i)33-s + (−0.587 + 0.809i)36-s + (−1.72 + 0.877i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.879 - 0.475i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.879 - 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3740346072\)
\(L(\frac12)\) \(\approx\) \(0.3740346072\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.743 - 0.669i)T \)
good2 \( 1 + (0.994 - 0.104i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.994 - 0.104i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-1.07 - 1.65i)T + (-0.406 + 0.913i)T^{2} \)
29 \( 1 + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (0.379 - 0.169i)T + (0.669 - 0.743i)T^{2} \)
37 \( 1 + (1.72 - 0.877i)T + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.104 + 0.994i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.483 + 0.185i)T + (0.743 - 0.669i)T^{2} \)
53 \( 1 + (1.07 + 0.170i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.895 + 0.994i)T + (-0.104 + 0.994i)T^{2} \)
61 \( 1 + (0.104 + 0.994i)T^{2} \)
67 \( 1 + (0.669 + 0.256i)T + (0.743 + 0.669i)T^{2} \)
71 \( 1 + (-0.873 - 1.20i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.207 + 0.978i)T^{2} \)
89 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.601 - 1.56i)T + (-0.743 + 0.669i)T^{2} \)
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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

−9.475639975760252525518188995903, −8.841841045599836938228754622222, −7.83038375900999326604977518961, −7.09103663752103809989798080933, −6.42388486460049007790708618322, −5.23097178256753606971380396139, −4.80031508817977760909364067125, −3.79392754987882984873193013073, −3.38432524514957773495591758081, −1.34397545706661861396930233542, 0.34796247024117681910921952118, 1.41164161481259750925904123865, 3.24682282797172278565610893940, 4.24203784044498925855324971487, 4.77650846126238704484370296876, 5.57610124468669227544170777456, 6.42742905851784212120760030224, 7.26805200965122652796902631448, 8.092431121892733698128471006432, 8.811062182766307582848446852262

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