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Properties

Label 2-15e3-75.29-c0-0-2
Degree $2$
Conductor $3375$
Sign $0.990 + 0.137i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $0.990 + 0.137i$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (2024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ 0.990 + 0.137i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.814753259\)
\(L(\frac12)\) \(\approx\) \(1.814753259\)
\(L(1)\) 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

−8.815986022271907191301223247750, −8.097557626405148496907089449777, −7.13815426204121128206874913953, −6.69968513588744557734505972401, −5.75104921929179846135189755015, −4.84288962609552941080537534787, −3.89023645500129368226229263369, −3.23193908112851736057354896403, −2.25252821613182854163250819550, −1.43160450307850129520232372989, 1.18712356863815812952797011067, 2.15155011242515581392475843215, 3.47778989169615192267272742265, 4.19562795238856637335805730037, 5.17573704514736112174493575727, 5.95486752473156564757581521308, 6.51463640767243350767248234583, 7.28372089157090954529625760987, 7.82376882045222861586065657412, 8.694963801476350562316918180464

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