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Properties

Label 2-1875-75.71-c0-0-3
Degree $2$
Conductor $1875$
Sign $0.844 - 0.535i$
Analytic cond. $0.935746$
Root an. cond. $0.967340$
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  + (0.587 + 0.190i)2-s + (−0.587 + 0.809i)3-s + (−0.5 − 0.363i)4-s + (−0.5 + 0.363i)6-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.587 − 0.190i)12-s + (0.951 + 1.30i)17-s − 0.618i·18-s + (0.5 − 0.363i)19-s + (1.53 + 0.5i)23-s + 24-s + (0.951 + 0.309i)27-s + (1.30 − 0.951i)31-s + i·32-s + ⋯
L(s)  = 1  + (0.587 + 0.190i)2-s + (−0.587 + 0.809i)3-s + (−0.5 − 0.363i)4-s + (−0.5 + 0.363i)6-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.587 − 0.190i)12-s + (0.951 + 1.30i)17-s − 0.618i·18-s + (0.5 − 0.363i)19-s + (1.53 + 0.5i)23-s + 24-s + (0.951 + 0.309i)27-s + (1.30 − 0.951i)31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102964694\)
\(L(\frac12)\) \(\approx\) \(1.102964694\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
good2 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−9.601137917293091790891770369262, −8.924965223175094357764289221915, −7.983785366183398033327942461543, −6.77384230945680924200282368790, −6.05235416239574908323667194938, −5.36307586871881680929557652099, −4.71507514817298473091225046890, −3.85348899967981939324967904835, −3.10252082115213211019250924275, −1.07099924914745953065240555302, 1.03380563186559177846250721650, 2.64055485185069724613858212341, 3.31925384718118450513417416969, 4.77605650797697560927659012653, 5.09168998750343521327133167260, 6.04900858092394957519348180113, 6.98413310011167736289564778744, 7.69689787751784093167377583774, 8.444618466965717562969584996691, 9.279563031435206673403227457693

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