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Properties

Label 2-375-15.2-c1-0-1
Degree $2$
Conductor $375$
Sign $-0.999 + 0.0182i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.548 + 0.548i)2-s + (−1.73 + 0.0316i)3-s + 1.39i·4-s + (0.932 − 0.966i)6-s + (1.30 + 1.30i)7-s + (−1.86 − 1.86i)8-s + (2.99 − 0.109i)9-s + 2.81i·11-s + (−0.0442 − 2.42i)12-s + (−3.20 + 3.20i)13-s − 1.43·14-s − 0.753·16-s + (0.175 − 0.175i)17-s + (−1.58 + 1.70i)18-s − 2.15i·19-s + ⋯
L(s)  = 1  + (−0.387 + 0.387i)2-s + (−0.999 + 0.0182i)3-s + 0.699i·4-s + (0.380 − 0.394i)6-s + (0.494 + 0.494i)7-s + (−0.658 − 0.658i)8-s + (0.999 − 0.0365i)9-s + 0.850i·11-s + (−0.0127 − 0.699i)12-s + (−0.888 + 0.888i)13-s − 0.383·14-s − 0.188·16-s + (0.0425 − 0.0425i)17-s + (−0.373 + 0.401i)18-s − 0.493i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.999 + 0.0182i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.999 + 0.0182i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00409279 - 0.447665i\)
\(L(\frac12)\) \(\approx\) \(0.00409279 - 0.447665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0316i)T \)
5 \( 1 \)
good2 \( 1 + (0.548 - 0.548i)T - 2iT^{2} \)
7 \( 1 + (-1.30 - 1.30i)T + 7iT^{2} \)
11 \( 1 - 2.81iT - 11T^{2} \)
13 \( 1 + (3.20 - 3.20i)T - 13iT^{2} \)
17 \( 1 + (-0.175 + 0.175i)T - 17iT^{2} \)
19 \( 1 + 2.15iT - 19T^{2} \)
23 \( 1 + (4.24 + 4.24i)T + 23iT^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
31 \( 1 + 6.12T + 31T^{2} \)
37 \( 1 + (-6.73 - 6.73i)T + 37iT^{2} \)
41 \( 1 - 3.05iT - 41T^{2} \)
43 \( 1 + (0.807 - 0.807i)T - 43iT^{2} \)
47 \( 1 + (6.38 - 6.38i)T - 47iT^{2} \)
53 \( 1 + (-9.01 - 9.01i)T + 53iT^{2} \)
59 \( 1 + 3.51T + 59T^{2} \)
61 \( 1 - 7.85T + 61T^{2} \)
67 \( 1 + (-1.34 - 1.34i)T + 67iT^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 + (-10.8 + 10.8i)T - 73iT^{2} \)
79 \( 1 - 6.52iT - 79T^{2} \)
83 \( 1 + (-0.146 - 0.146i)T + 83iT^{2} \)
89 \( 1 - 0.309T + 89T^{2} \)
97 \( 1 + (6.77 + 6.77i)T + 97iT^{2} \)
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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

−11.85300211327401735450692620416, −11.13677821692003679498555474081, −9.804924996298133860811870736435, −9.196667018225868277961601209238, −7.895430747617281841189982743561, −7.16273874611138890650142137056, −6.29492973028872686911763621913, −5.00503571965225468648255597718, −4.10419792836871419506183301608, −2.16028072044720482791509653939, 0.37416438706007661388986098015, 1.86664501541885132731469763470, 3.86375644412081998438374960946, 5.42489298798715473352668569029, 5.67712289199087435340985857943, 7.13421413691059358744100819179, 8.073022608720199465228780109234, 9.484177387573772015575674597238, 10.12730566795325202497600471111, 11.04028664656723526501183366467

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