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Properties

Label 2-675-5.3-c0-0-2
Degree $2$
Conductor $675$
Sign $0.945 - 0.326i$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
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  + i·4-s + (1.22 − 1.22i)7-s − 16-s + i·19-s + (1.22 + 1.22i)28-s + 31-s + (−1.22 + 1.22i)37-s + (−1.22 − 1.22i)43-s − 1.99i·49-s − 61-s − i·64-s + (−1.22 − 1.22i)73-s − 76-s − i·79-s + (−1.22 + 1.22i)97-s + ⋯
L(s)  = 1  + i·4-s + (1.22 − 1.22i)7-s − 16-s + i·19-s + (1.22 + 1.22i)28-s + 31-s + (−1.22 + 1.22i)37-s + (−1.22 − 1.22i)43-s − 1.99i·49-s − 61-s − i·64-s + (−1.22 − 1.22i)73-s − 76-s − i·79-s + (−1.22 + 1.22i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ 0.945 - 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034918685\)
\(L(\frac12)\) \(\approx\) \(1.034918685\)
\(L(1)\) 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 \( 1 - iT^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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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

−10.72382500755232839908505086296, −10.10456985609110112495793942794, −8.740921047563922360928856367337, −8.066018408074557773732614864473, −7.45057385091042767982749692910, −6.55810132509333050460831109417, −5.03736764589016524922479413453, −4.22513977990903072098060814147, −3.30526644552585761737677076977, −1.68777958372204334564742477948, 1.59296781081227783987814082373, 2.66337317044612063450904882123, 4.54166633603933565975712175367, 5.22059484773257747419478587281, 6.00988172258741036538284739376, 7.06145238520667371942355739240, 8.285837107122008038410596797855, 8.924231481404348483644161427424, 9.762127944616193245298331333996, 10.78071287903503856332010540663

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