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Properties

Label 2-1875-75.14-c0-0-2
Degree $2$
Conductor $1875$
Sign $0.187 - 0.982i$
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 + 0.309i)3-s + (−0.309 + 0.951i)4-s − 0.618i·7-s + (0.809 + 0.587i)9-s + (−0.587 + 0.809i)12-s + (−0.951 + 1.30i)13-s + (−0.809 − 0.587i)16-s + (0.5 + 1.53i)19-s + (0.190 − 0.587i)21-s + (0.587 + 0.809i)27-s + (0.587 + 0.190i)28-s + (0.190 + 0.587i)31-s + (−0.809 + 0.587i)36-s + (0.951 − 1.30i)37-s + (−1.30 + 0.951i)39-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s − 0.618i·7-s + (0.809 + 0.587i)9-s + (−0.587 + 0.809i)12-s + (−0.951 + 1.30i)13-s + (−0.809 − 0.587i)16-s + (0.5 + 1.53i)19-s + (0.190 − 0.587i)21-s + (0.587 + 0.809i)27-s + (0.587 + 0.190i)28-s + (0.190 + 0.587i)31-s + (−0.809 + 0.587i)36-s + (0.951 − 1.30i)37-s + (−1.30 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.417228353\)
\(L(\frac12)\) \(\approx\) \(1.417228353\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
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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

−9.459416678994646751526521436288, −8.838411815259698371753712725226, −8.000563425247089160577125240984, −7.40421184417133516174815480635, −6.83846155248142874275191159638, −5.34851798528894669378990704091, −4.21919908043872246226886448492, −3.95285640909954738143045839996, −2.88034072340369845206847607166, −1.85845530638469205106969670978, 1.00047442226764217397469606939, 2.41614666596351768715712621358, 2.99568690512544094006431687872, 4.45107850250516583020678614856, 5.13965744037504551641257823664, 6.05907757957149884131240303644, 6.92621348982467059063758350504, 7.81155907874941823293851253788, 8.488196573304206773347598690656, 9.458412863861730575808292478742

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