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Properties

Label 2-15e3-75.56-c0-0-0
Degree $2$
Conductor $3375$
Sign $0.770 - 0.637i$
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  + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s − 0.618·7-s + (−0.587 + 0.809i)8-s + (0.951 − 0.309i)11-s + (−0.363 + 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.5 − 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.309 − 0.224i)28-s + (0.951 + 1.30i)29-s − i·32-s + (0.190 − 0.587i)34-s + (−0.5 + 1.53i)37-s + (0.587 + 0.190i)38-s + ⋯
L(s)  = 1  + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s − 0.618·7-s + (−0.587 + 0.809i)8-s + (0.951 − 0.309i)11-s + (−0.363 + 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.5 − 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.309 − 0.224i)28-s + (0.951 + 1.30i)29-s − i·32-s + (0.190 − 0.587i)34-s + (−0.5 + 1.53i)37-s + (0.587 + 0.190i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380715329\)
\(L(\frac12)\) \(\approx\) \(1.380715329\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.914709093613294634087650922584, −8.177951826718702642677008181575, −7.41148149961304476251936588104, −6.49580441405776329845077131596, −5.76970952376119644712926847724, −4.99746651758078235522363372333, −4.14045904344825707025925792716, −3.36247672170627758410762225312, −2.82570393338442623337829796625, −1.23797899058232323698002500727, 0.810738413735601281582788373788, 2.23231532146518522368893779632, 3.55458023567640894416274516656, 3.98559631915088052707009646969, 4.87235673141337868489788547633, 5.80243054361561462943397091587, 6.25858317941318452388920783453, 7.02526505541967281770161235613, 7.926915384663545089319119093808, 8.897361276246088711709662470295

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