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Properties

Label 2-15e3-5.3-c0-0-1
Degree $2$
Conductor $3375$
Sign $-i$
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  + (0.294 − 0.294i)2-s + 0.827i·4-s + (0.537 + 0.537i)8-s − 0.511·16-s + (−1.40 + 1.40i)17-s + 1.95i·19-s + (−0.575 − 0.575i)23-s + 0.209·31-s + (−0.687 + 0.687i)32-s + 0.827i·34-s + (0.575 + 0.575i)38-s − 0.338·46-s + (1.34 − 1.34i)47-s + i·49-s + (−1.05 − 1.05i)53-s + ⋯
L(s)  = 1  + (0.294 − 0.294i)2-s + 0.827i·4-s + (0.537 + 0.537i)8-s − 0.511·16-s + (−1.40 + 1.40i)17-s + 1.95i·19-s + (−0.575 − 0.575i)23-s + 0.209·31-s + (−0.687 + 0.687i)32-s + 0.827i·34-s + (0.575 + 0.575i)38-s − 0.338·46-s + (1.34 − 1.34i)47-s + i·49-s + (−1.05 − 1.05i)53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3375\)    =    \(3^{3} \cdot 5^{3}\)
Sign: $-i$
Analytic conductor: \(1.68434\)
Root analytic conductor: \(1.29782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3375} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3375,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224718929\)
\(L(\frac12)\) \(\approx\) \(1.224718929\)
\(L(1)\) 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 \( 1 + (-0.294 + 0.294i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
19 \( 1 - 1.95iT - T^{2} \)
23 \( 1 + (0.575 + 0.575i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.209T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
53 \( 1 + (1.05 + 1.05i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.33T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.33iT - T^{2} \)
83 \( 1 + (-1.05 - 1.05i)T + iT^{2} \)
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

−8.680854324969767095023370392021, −8.314161831096749696420454923385, −7.65786067878678106258630477190, −6.67638536658535259411218878661, −6.05168687772614425918688706579, −5.04224705395428698246140522028, −3.91741784626914654898473033472, −3.85856135652336802400570444076, −2.48614761004840701738026049128, −1.76382045518483821489354458043, 0.63336455953677212041394277839, 2.06032820121924514978865649224, 2.91547532187192105374833048489, 4.31316866300042303763116317771, 4.76678467395587837176688660790, 5.53592805072953991787094459530, 6.43111121857085257104541909020, 6.98230875980017954172741403706, 7.60997433294290453352252701708, 8.879015680680507224367487762270

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