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Properties

Label 2-1875-75.11-c0-0-1
Degree $2$
Conductor $1875$
Sign $0.684 + 0.728i$
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  + (−0.951 − 1.30i)2-s + (0.951 + 0.309i)3-s + (−0.500 + 1.53i)4-s + (−0.499 − 1.53i)6-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.951 + 1.30i)12-s + (0.587 − 0.190i)17-s − 1.61i·18-s + (0.5 + 1.53i)19-s + (−0.363 − 0.5i)23-s + 24-s + (0.587 + 0.809i)27-s + (0.190 + 0.587i)31-s − 0.999i·32-s + ⋯
L(s)  = 1  + (−0.951 − 1.30i)2-s + (0.951 + 0.309i)3-s + (−0.500 + 1.53i)4-s + (−0.499 − 1.53i)6-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.951 + 1.30i)12-s + (0.587 − 0.190i)17-s − 1.61i·18-s + (0.5 + 1.53i)19-s + (−0.363 − 0.5i)23-s + 24-s + (0.587 + 0.809i)27-s + (0.190 + 0.587i)31-s − 0.999i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9787041563\)
\(L(\frac12)\) \(\approx\) \(0.9787041563\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.487886395140085966512293896341, −8.735947682805065710347848936821, −8.029956050951639331060224092633, −7.55857960958403834384967141699, −6.20831645930484803152358701580, −4.95497634067610214431082560623, −3.79575818001261710634178592583, −3.25451192994799804004641519055, −2.26115412936775661143143234112, −1.34945360190227659040966974479, 1.06739696846825399963147358720, 2.52681859802802204093871962869, 3.64115340243626272193025896195, 4.87255743997650686717735644692, 5.84598185264258248688549756509, 6.71787046354028690819684935251, 7.32045370475239576400867217516, 7.942130705964669085736342321751, 8.569224990635685517379326155461, 9.380340676316936669795876504788

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