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Properties

Label 2-15e3-5.2-c0-0-2
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  + (1.40 + 1.40i)2-s + 2.95i·4-s + (−2.75 + 2.75i)8-s − 4.78·16-s + (1.05 + 1.05i)17-s − 0.209i·19-s + (−0.294 + 0.294i)23-s − 1.33·31-s + (−3.97 − 3.97i)32-s + 2.95i·34-s + (0.294 − 0.294i)38-s − 0.827·46-s + (0.831 + 0.831i)47-s − i·49-s + (0.575 − 0.575i)53-s + ⋯
L(s)  = 1  + (1.40 + 1.40i)2-s + 2.95i·4-s + (−2.75 + 2.75i)8-s − 4.78·16-s + (1.05 + 1.05i)17-s − 0.209i·19-s + (−0.294 + 0.294i)23-s − 1.33·31-s + (−3.97 − 3.97i)32-s + 2.95i·34-s + (0.294 − 0.294i)38-s − 0.827·46-s + (0.831 + 0.831i)47-s − i·49-s + (0.575 − 0.575i)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

\(L(\frac{1}{2})\) \(\approx\) \(2.444870660\)
\(L(\frac12)\) \(\approx\) \(2.444870660\)
\(L(1)\) 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 \( 1 + (-1.40 - 1.40i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.05 - 1.05i)T + iT^{2} \)
19 \( 1 + 0.209iT - T^{2} \)
23 \( 1 + (0.294 - 0.294i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.33T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.831 - 0.831i)T + iT^{2} \)
53 \( 1 + (-0.575 + 0.575i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.82T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.82iT - T^{2} \)
83 \( 1 + (0.575 - 0.575i)T - iT^{2} \)
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

−8.707248398435790259416492649657, −8.167382709151010676479120769124, −7.42190366085116364974357928470, −6.89574925692158753917385695927, −5.92064490874512054635908200962, −5.59612685501772520913413410014, −4.72479917295082360887864063188, −3.83177943084697839795209948799, −3.35296993191824917866157549070, −2.15468120209287805675107230215, 0.922265858554594501490798561337, 2.05259722822569922707255922239, 2.87889391208419212531473712069, 3.67039222024123940723395906997, 4.35881016260369444063880672141, 5.34453251766047044615205563134, 5.64041006078442877422895961344, 6.66171079726048173470960738635, 7.45091301345448680029434584173, 8.767623984785994925017958332421

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