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Properties

Label 2-1875-75.29-c0-0-6
Degree $2$
Conductor $1875$
Sign $-0.0627 + 0.998i$
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  + (0.5 − 1.53i)2-s + (0.809 + 0.587i)3-s + (−1.30 − 0.951i)4-s + (1.30 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.5 − 1.53i)12-s + (0.5 − 0.363i)17-s + 1.61·18-s + (1.30 − 0.951i)19-s + (−0.190 + 0.587i)23-s − 24-s + (−0.309 + 0.951i)27-s + (−0.5 + 0.363i)31-s − 0.999·32-s + ⋯
L(s)  = 1  + (0.5 − 1.53i)2-s + (0.809 + 0.587i)3-s + (−1.30 − 0.951i)4-s + (1.30 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.5 − 1.53i)12-s + (0.5 − 0.363i)17-s + 1.61·18-s + (1.30 − 0.951i)19-s + (−0.190 + 0.587i)23-s − 24-s + (−0.309 + 0.951i)27-s + (−0.5 + 0.363i)31-s − 0.999·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-0.0627 + 0.998i$
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: \((2,\ 1875,\ (\ :0),\ -0.0627 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.900111752\)
\(L(\frac12)\) \(\approx\) \(1.900111752\)
\(L(1)\) 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 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
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

−9.527887615575562623050569667638, −8.846761728730680887657007667721, −7.80250693762180390817761089951, −7.01651318768139102582643655229, −5.39696348152004500387417065680, −4.91479204823129170446540949600, −3.88476055417383479012814432481, −3.24102776589057384849574151790, −2.50933943588535335170919947996, −1.38141177586643142650613330536, 1.62203946720057198068569484359, 3.11241577117848876110238413106, 3.92719782597564167865218862184, 4.93136844251792157598091555221, 5.92457854714881462487742387979, 6.41324468424209653638089570182, 7.45912916691355715920801218333, 7.74554867779985242237693548741, 8.465512202168947769284315732128, 9.281809171583713671585289696365

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