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Properties

Label 2-15e3-75.44-c0-0-0
Degree $2$
Conductor $3375$
Sign $0.320 - 0.947i$
Analytic cond. $1.68434$
Root an. cond. $1.29782$
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  + (0.190 + 0.587i)2-s + (0.5 − 0.363i)4-s + 0.618i·7-s + (0.809 + 0.587i)8-s + (−0.951 + 0.309i)11-s + (−0.363 + 0.118i)14-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.363 − 0.5i)22-s + (0.309 + 0.951i)23-s + (0.224 + 0.309i)28-s + (0.951 + 1.30i)29-s + 32-s + (−0.190 + 0.587i)34-s + (1.53 + 0.5i)37-s + (0.190 − 0.587i)38-s + ⋯
L(s)  = 1  + (0.190 + 0.587i)2-s + (0.5 − 0.363i)4-s + 0.618i·7-s + (0.809 + 0.587i)8-s + (−0.951 + 0.309i)11-s + (−0.363 + 0.118i)14-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.363 − 0.5i)22-s + (0.309 + 0.951i)23-s + (0.224 + 0.309i)28-s + (0.951 + 1.30i)29-s + 32-s + (−0.190 + 0.587i)34-s + (1.53 + 0.5i)37-s + (0.190 − 0.587i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ 0.320 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.631362195\)
\(L(\frac12)\) \(\approx\) \(1.631362195\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - 0.618iT - T^{2} \)
11 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
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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

−8.779455198778288933289456036959, −7.999011930531692521959593305482, −7.48702735847800570897272130650, −6.56351193323408598477634608523, −6.03641098568991254295231926857, −5.13985946833394434742488670653, −4.76463254907688600639217961684, −3.32263197381145296044167395040, −2.45983679023582858205441624199, −1.48817135700339428909166001655, 0.957266787299477945118446042367, 2.32756874442080653055068658116, 2.91263867614615838696680124818, 3.91692490495426073271954310628, 4.55644609684162873163226294190, 5.60446210235564866438013295887, 6.50131195626895015674600318099, 7.18651069845510567650397778820, 8.015759447806913406056071168359, 8.344236159268679098718742902432

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