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Properties

Label 2-15e3-75.14-c0-0-2
Degree $2$
Conductor $3375$
Sign $0.909 + 0.414i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $0.909 + 0.414i$
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: \((2,\ 3375,\ (\ :0),\ 0.909 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.637131075\)
\(L(\frac12)\) \(\approx\) \(2.637131075\)
\(L(1)\) 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

−8.974052466238614582201886331066, −8.028695857000165133371820197895, −7.07621637296528154774963736524, −5.97868660512359079383362074421, −5.64075840308078484124770231517, −4.83160167623526757927342941492, −4.03729197954852946667498575555, −3.19532386369128127487132858398, −2.27886050743633891207728740695, −1.74610471764296551370592487421, 1.11800821955837560004966233250, 2.82323488685608588874971069670, 3.89046957963771416675211094478, 4.09272777441989001290999083632, 4.98258768768573571335912201900, 5.95738404508630309641885392840, 6.51265966775642593045193737573, 7.12033237285711298428550114393, 7.80852946498905808619777249272, 8.495088386169649637889559471693

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