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Properties

Label 2-15e3-75.14-c0-0-3
Degree $2$
Conductor $3375$
Sign $-0.778 + 0.628i$
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  + (1.30 − 0.951i)2-s + (0.500 − 1.53i)4-s − 1.61i·7-s + (−0.309 − 0.951i)8-s + (−0.587 − 0.809i)11-s + (−1.53 − 2.11i)14-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (−1.53 − 0.5i)22-s + (−0.809 + 0.587i)23-s + (−2.48 − 0.809i)28-s + (0.587 + 0.190i)29-s + 0.999·32-s + (−1.30 − 0.951i)34-s + (−0.363 + 0.5i)37-s + (1.30 + 0.951i)38-s + ⋯
L(s)  = 1  + (1.30 − 0.951i)2-s + (0.500 − 1.53i)4-s − 1.61i·7-s + (−0.309 − 0.951i)8-s + (−0.587 − 0.809i)11-s + (−1.53 − 2.11i)14-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (−1.53 − 0.5i)22-s + (−0.809 + 0.587i)23-s + (−2.48 − 0.809i)28-s + (0.587 + 0.190i)29-s + 0.999·32-s + (−1.30 − 0.951i)34-s + (−0.363 + 0.5i)37-s + (1.30 + 0.951i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $-0.778 + 0.628i$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (2699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ -0.778 + 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.419063170\)
\(L(\frac12)\) \(\approx\) \(2.419063170\)
\(L(1)\) 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 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + 1.61iT - T^{2} \)
11 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
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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

−8.331185840800824741174097599422, −7.74249109664711751024737939440, −6.82316548872570696532099590126, −6.04025494503956566810538714950, −5.13145357245664530275868175228, −4.57595332421653518577634416936, −3.58661163425654652855516621708, −3.29946550271177079113300853197, −2.06548715048835269961423851834, −0.939563365895353182765083122701, 2.14043681931453855391218729653, 2.77779072323280197233397491131, 3.91066280422114059679011119066, 4.69759459470990334396276912055, 5.39627006665148752725824449148, 5.89334612308718151576613009167, 6.67814902549947154740513827509, 7.32315306406830239329341446890, 8.288549179653818121371374766224, 8.717936988954176964515470285225

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