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Properties

Label 2-375-5.4-c1-0-3
Degree $2$
Conductor $375$
Sign $-1$
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  + 1.87i·2-s + i·3-s − 1.51·4-s − 1.87·6-s + 3.55i·7-s + 0.901i·8-s − 9-s + 4.51·11-s − 1.51i·12-s − 1.68i·13-s − 6.66·14-s − 4.73·16-s − 3.85i·17-s − 1.87i·18-s − 6.13·19-s + ⋯
L(s)  = 1  + 1.32i·2-s + 0.577i·3-s − 0.759·4-s − 0.765·6-s + 1.34i·7-s + 0.318i·8-s − 0.333·9-s + 1.36·11-s − 0.438i·12-s − 0.466i·13-s − 1.78·14-s − 1.18·16-s − 0.934i·17-s − 0.442i·18-s − 1.40·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(-1.34323i\)
\(L(\frac12)\) \(\approx\) \(-1.34323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
good2 \( 1 - 1.87iT - 2T^{2} \)
7 \( 1 - 3.55iT - 7T^{2} \)
11 \( 1 - 4.51T + 11T^{2} \)
13 \( 1 + 1.68iT - 13T^{2} \)
17 \( 1 + 3.85iT - 17T^{2} \)
19 \( 1 + 6.13T + 19T^{2} \)
23 \( 1 - 3.57iT - 23T^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 + 6.54iT - 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 9.77iT - 43T^{2} \)
47 \( 1 + 4.13iT - 47T^{2} \)
53 \( 1 - 3.49iT - 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 8.72T + 61T^{2} \)
67 \( 1 - 9.94iT - 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 5.75iT - 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 - 5.66iT - 83T^{2} \)
89 \( 1 + 4.12T + 89T^{2} \)
97 \( 1 - 5.91iT - 97T^{2} \)
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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

−11.75372204437480739614215866549, −11.02759684646935743901436267451, −9.516265856024358788051667983460, −8.960949112596700032779921724476, −8.182100325370538681922012976594, −6.95906479487194391613775128029, −6.04294930429454891371239332152, −5.35051100230771050920860148349, −4.17134390110150705919403294782, −2.49324380308616866714922418345, 0.964328604534761658639698097555, 2.11858504209262853722309557137, 3.75673191286565757675778590039, 4.31124514978421649190338603238, 6.43560442384215486352495769580, 6.91935258241873846154145144895, 8.319463830777173072041049539178, 9.298153430494213217030429996249, 10.39204130065158161122521335825, 10.88804326589365859580304652306

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