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Properties

Label 2-15e3-5.2-c0-0-6
Degree $2$
Conductor $3375$
Sign $1$
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  + (0.575 + 0.575i)2-s − 0.338i·4-s + (0.769 − 0.769i)8-s + 0.547·16-s + (−0.294 − 0.294i)17-s + 1.82i·19-s + (1.05 − 1.05i)23-s + 1.95·31-s + (−0.454 − 0.454i)32-s − 0.338i·34-s + (−1.05 + 1.05i)38-s + 1.20·46-s + (−0.831 − 0.831i)47-s − i·49-s + (1.40 − 1.40i)53-s + ⋯
L(s)  = 1  + (0.575 + 0.575i)2-s − 0.338i·4-s + (0.769 − 0.769i)8-s + 0.547·16-s + (−0.294 − 0.294i)17-s + 1.82i·19-s + (1.05 − 1.05i)23-s + 1.95·31-s + (−0.454 − 0.454i)32-s − 0.338i·34-s + (−1.05 + 1.05i)38-s + 1.20·46-s + (−0.831 − 0.831i)47-s − i·49-s + (1.40 − 1.40i)53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (1432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.826829594\)
\(L(\frac12)\) \(\approx\) \(1.826829594\)
\(L(1)\) 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 \( 1 + (-0.575 - 0.575i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.294 + 0.294i)T + iT^{2} \)
19 \( 1 - 1.82iT - T^{2} \)
23 \( 1 + (-1.05 + 1.05i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.95T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
53 \( 1 + (-1.40 + 1.40i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.209T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 0.209iT - T^{2} \)
83 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.556634734224950099515989382788, −8.065090965880050197279227878881, −6.97336468668997464891151143519, −6.59107060563373037456526182697, −5.75188174701808866489485062547, −5.06969442458236443502704412888, −4.33371931744678430725110989130, −3.50031748641368130697923041717, −2.28476099868144884588710287688, −1.07483976215468366103946882184, 1.33118215215135597348859513116, 2.66608315167428115244170715612, 3.03235953501301402068837527928, 4.24407399085825798197818170072, 4.69852961859277240493314481691, 5.56560090620033309344972525398, 6.62454082140798058312555942750, 7.29462756153644745897848169147, 8.048890784684019988378233626634, 8.841181712204664889512552326659

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