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Properties

Label 2-1875-75.41-c0-0-6
Degree $2$
Conductor $1875$
Sign $0.904 + 0.425i$
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  + (0.363 − 0.5i)2-s + (0.951 − 0.309i)3-s + (0.190 + 0.587i)4-s + (0.190 − 0.587i)6-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.363 + 0.5i)12-s + (−1.53 − 0.5i)17-s − 0.618i·18-s + (−0.190 + 0.587i)19-s + (0.951 − 1.30i)23-s + 24-s + (0.587 − 0.809i)27-s + (−0.5 + 1.53i)31-s + i·32-s + ⋯
L(s)  = 1  + (0.363 − 0.5i)2-s + (0.951 − 0.309i)3-s + (0.190 + 0.587i)4-s + (0.190 − 0.587i)6-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + (0.363 + 0.5i)12-s + (−1.53 − 0.5i)17-s − 0.618i·18-s + (−0.190 + 0.587i)19-s + (0.951 − 1.30i)23-s + 24-s + (0.587 − 0.809i)27-s + (−0.5 + 1.53i)31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.904 + 0.425i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.904 + 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.052448868\)
\(L(\frac12)\) \(\approx\) \(2.052448868\)
\(L(1)\) 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 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
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

−9.099205108312436445633881112318, −8.629783728230845107727777917678, −7.895339885219633902782850273702, −6.98346900284688691268571065738, −6.55513807257695208537686628107, −4.92413037582424366676225084669, −4.25288688716514084208717167145, −3.28427235302142754123838414120, −2.58362880719824324217359002829, −1.66258266450929655318734817430, 1.62374622446136079672573915125, 2.57080871569358104443564984785, 3.82871141255607362239156896579, 4.56633701518613567774164127636, 5.36556914411977283585988190692, 6.38472512926495616570785952092, 7.10558073442618328835399972063, 7.78947838997579116526222335864, 8.771723159334716527125820079078, 9.390770008550226076073350933955

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