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Properties

Label 2-675-75.11-c0-0-0
Degree $2$
Conductor $675$
Sign $-0.876 - 0.481i$
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  + (0.951 + 1.30i)2-s + (−0.500 + 1.53i)4-s + (−0.587 + 0.809i)5-s − 1.61·7-s + (−0.951 + 0.309i)8-s − 1.61·10-s + (0.587 + 0.809i)11-s + (−1.53 − 2.11i)14-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.951 − 1.30i)20-s + (−0.5 + 1.53i)22-s + (0.587 + 0.809i)23-s + (−0.309 − 0.951i)25-s + (0.809 − 2.48i)28-s + (0.587 + 0.190i)29-s + ⋯
L(s)  = 1  + (0.951 + 1.30i)2-s + (−0.500 + 1.53i)4-s + (−0.587 + 0.809i)5-s − 1.61·7-s + (−0.951 + 0.309i)8-s − 1.61·10-s + (0.587 + 0.809i)11-s + (−1.53 − 2.11i)14-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.951 − 1.30i)20-s + (−0.5 + 1.53i)22-s + (0.587 + 0.809i)23-s + (−0.309 − 0.951i)25-s + (0.809 − 2.48i)28-s + (0.587 + 0.190i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.200019928\)
\(L(\frac12)\) \(\approx\) \(1.200019928\)
\(L(1)\) 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 \( 1 + (0.587 - 0.809i)T \)
good2 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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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

−11.20359909049060464921932385798, −10.03194466176241139302816600583, −9.329431062246717558064059835790, −8.037823115787514228329595153694, −7.04394374285702898572798004763, −6.80353434566048179118539485743, −5.93371942188501411469829592658, −4.75250963630345011949499416814, −3.70199253073491333833379349343, −2.99529557305337859936780823234, 1.10082940758913253780550383607, 2.91195562136286307710062105242, 3.62459098667362633305083208672, 4.40206196540362936781975138971, 5.64909821977413443190143460781, 6.39094522465226676647919914606, 7.85579614834427161845309067125, 8.926221297085185185961124086780, 9.761630548703082711737802406047, 10.46079922482792859708127837273

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