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Properties

Label 2-1875-75.29-c0-0-2
Degree $2$
Conductor $1875$
Sign $-0.535 - 0.844i$
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  + (−0.190 + 0.587i)2-s + (0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (−0.5 + 0.363i)6-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.190 + 0.587i)12-s + (−1.30 + 0.951i)17-s − 0.618·18-s + (−0.5 + 0.363i)19-s + (0.5 − 1.53i)23-s − 24-s + (−0.309 + 0.951i)27-s + (1.30 − 0.951i)31-s − 32-s + ⋯
L(s)  = 1  + (−0.190 + 0.587i)2-s + (0.809 + 0.587i)3-s + (0.5 + 0.363i)4-s + (−0.5 + 0.363i)6-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.190 + 0.587i)12-s + (−1.30 + 0.951i)17-s − 0.618·18-s + (−0.5 + 0.363i)19-s + (0.5 − 1.53i)23-s − 24-s + (−0.309 + 0.951i)27-s + (1.30 − 0.951i)31-s − 32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.467961206\)
\(L(\frac12)\) \(\approx\) \(1.467961206\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 \)
good2 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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

−9.445277486851412789258634519358, −8.530700928964905932768420695429, −8.405141380337953271029333653903, −7.42850448946980612426228021367, −6.60204507992941124793154420054, −5.91899267412486467212416111912, −4.62370902658069045407526558097, −3.96202175540882700259162257822, −2.77730385222266412627494783171, −2.12073974985003143064213835291, 1.06997600884240014614772654987, 2.21503998564365411583514453905, 2.85022446296130981873765571046, 3.85347898939405790315760959161, 5.06806790988792131388811957971, 6.21818180084769290830130199641, 6.91559909608791205661189612948, 7.44819607219922384310925090405, 8.644689535371245192831261597501, 9.107805284092335082630151779365

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