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Properties

Label 2-375-5.4-c1-0-11
Degree $2$
Conductor $375$
Sign $i$
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  − 0.618i·2-s − i·3-s + 1.61·4-s − 0.618·6-s − 0.618i·7-s − 2.23i·8-s − 9-s − 0.236·11-s − 1.61i·12-s − 6.23i·13-s − 0.381·14-s + 1.85·16-s + 6.61i·17-s + 0.618i·18-s + 5·19-s + ⋯
L(s)  = 1  − 0.437i·2-s − 0.577i·3-s + 0.809·4-s − 0.252·6-s − 0.233i·7-s − 0.790i·8-s − 0.333·9-s − 0.0711·11-s − 0.467i·12-s − 1.72i·13-s − 0.102·14-s + 0.463·16-s + 1.60i·17-s + 0.145i·18-s + 1.14·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $i$
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: \((2,\ 375,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13849 - 1.13849i\)
\(L(\frac12)\) \(\approx\) \(1.13849 - 1.13849i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618iT - 2T^{2} \)
7 \( 1 + 0.618iT - 7T^{2} \)
11 \( 1 + 0.236T + 11T^{2} \)
13 \( 1 + 6.23iT - 13T^{2} \)
17 \( 1 - 6.61iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + 3.47iT - 23T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 + 2.14T + 31T^{2} \)
37 \( 1 - 0.236iT - 37T^{2} \)
41 \( 1 - 5.09T + 41T^{2} \)
43 \( 1 - 8.56iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 4.09iT - 53T^{2} \)
59 \( 1 + 8.61T + 59T^{2} \)
61 \( 1 + 6.61T + 61T^{2} \)
67 \( 1 - 11.4iT - 67T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 + 4.85iT - 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 - 18.0iT - 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−10.94646648884909691420723441034, −10.61931205109921944011693048865, −9.491932569640498671310839694899, −8.027293023865392210903541812835, −7.55091059897399862744243262354, −6.32845037828375589503866646329, −5.53791015238192404014654171644, −3.70986467532341963545450155992, −2.65200531292813349579515280430, −1.19498675371726905732334695798, 2.08497608240601334452888285033, 3.44992727613305518495162110329, 4.90447248257941197319622042348, 5.79340181590649843956732283093, 6.99552446476749468614579150637, 7.59454254658715931410736884976, 9.075288696536675703713381941401, 9.531582958660789429757118743659, 10.87041625623264250256102749932, 11.59520351050514566905034165104

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