Properties

Label 2-1875-5.4-c1-0-78
Degree $2$
Conductor $1875$
Sign $-i$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
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  − 1.70i·2-s i·3-s − 0.911·4-s − 1.70·6-s − 3.94i·7-s − 1.85i·8-s − 9-s − 5.90·11-s + 0.911i·12-s − 3.29i·13-s − 6.72·14-s − 4.99·16-s + 2.70i·17-s + 1.70i·18-s + 2.35·19-s + ⋯
L(s)  = 1  − 1.20i·2-s − 0.577i·3-s − 0.455·4-s − 0.696·6-s − 1.49i·7-s − 0.656i·8-s − 0.333·9-s − 1.78·11-s + 0.263i·12-s − 0.912i·13-s − 1.79·14-s − 1.24·16-s + 0.656i·17-s + 0.402i·18-s + 0.540·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-i$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.029830013\)
\(L(\frac12)\) \(\approx\) \(1.029830013\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.70iT - 2T^{2} \)
7 \( 1 + 3.94iT - 7T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
13 \( 1 + 3.29iT - 13T^{2} \)
17 \( 1 - 2.70iT - 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
23 \( 1 - 0.584iT - 23T^{2} \)
29 \( 1 - 3.91T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 + 0.0208iT - 37T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 + 1.27iT - 43T^{2} \)
47 \( 1 - 5.43iT - 47T^{2} \)
53 \( 1 + 2.81iT - 53T^{2} \)
59 \( 1 - 4.69T + 59T^{2} \)
61 \( 1 - 5.58T + 61T^{2} \)
67 \( 1 - 6.03iT - 67T^{2} \)
71 \( 1 + 8.10T + 71T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + 0.781iT - 83T^{2} \)
89 \( 1 - 3.47T + 89T^{2} \)
97 \( 1 + 2.45iT - 97T^{2} \)
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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

−8.600563103618325770652623771252, −7.72286800422401649942455735317, −7.33448980210188977763656302440, −6.30858798827037606344691283074, −5.23867754778067747269709306492, −4.24447776660971490798713210906, −3.24543832825035008740385130074, −2.58216258668697336337486227534, −1.31923020316798025819087136023, −0.37286491153632865127854876810, 2.31379087394228341437742462463, 2.88084088527466749884597029074, 4.53749526585712180250888225470, 5.27296490435453282417280577310, 5.66766871931553499956708648310, 6.60423366669396616451279753297, 7.46773837328986912189323057575, 8.293090058394282574475866305956, 8.769924575548593532033353766002, 9.580418599616956165515126863166

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