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Properties

Label 2-375-3.2-c0-0-3
Degree $2$
Conductor $375$
Sign $-1$
Analytic cond. $0.187149$
Root an. cond. $0.432607$
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  − 1.61i·2-s − i·3-s − 1.61·4-s − 1.61·6-s + i·8-s − 9-s + 1.61i·12-s + 0.618i·17-s + 1.61i·18-s + 1.61·19-s − 0.618i·23-s + 24-s + i·27-s + 0.618·31-s + i·32-s + ⋯
L(s)  = 1  − 1.61i·2-s − i·3-s − 1.61·4-s − 1.61·6-s + i·8-s − 9-s + 1.61i·12-s + 0.618i·17-s + 1.61i·18-s + 1.61·19-s − 0.618i·23-s + 24-s + i·27-s + 0.618·31-s + i·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(0.187149\)
Root analytic conductor: \(0.432607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7490433539\)
\(L(\frac12)\) \(\approx\) \(0.7490433539\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
good2 \( 1 + 1.61iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 0.618iT - T^{2} \)
19 \( 1 - 1.61T + T^{2} \)
23 \( 1 + 0.618iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.61iT - T^{2} \)
53 \( 1 - 1.61iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.61T + T^{2} \)
83 \( 1 - 1.61iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.32404863252346327906680881076, −10.47919340170411193090495192234, −9.503801470591676704005465095242, −8.576881687169636642026612038763, −7.53549687578472053496027553107, −6.33860307761616180470448134622, −5.02336156175189142139551065295, −3.55919105250023890180476697494, −2.52522144783585188034381875368, −1.24918510293156742746126274530, 3.21035856491920353565554623693, 4.60977840084132927145488570323, 5.33010878118530730713846032727, 6.24850594713023597855250916066, 7.40951472385205795234527931908, 8.190919204605703904783814529545, 9.276211560170533547559692964246, 9.744865653868953612419314283608, 11.07705708833112441797024160273, 11.92649565707836282489460152812

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