Properties

Label 2-975-5.4-c1-0-28
Degree $2$
Conductor $975$
Sign $0.894 + 0.447i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + 2.08i·2-s + i·3-s − 2.35·4-s − 2.08·6-s − 4.08i·7-s − 0.734i·8-s − 9-s − 3.43·11-s − 2.35i·12-s + i·13-s + 8.52·14-s − 3.17·16-s − 7.17i·17-s − 2.08i·18-s − 7.52·19-s + ⋯
L(s)  = 1  + 1.47i·2-s + 0.577i·3-s − 1.17·4-s − 0.851·6-s − 1.54i·7-s − 0.259i·8-s − 0.333·9-s − 1.03·11-s − 0.678i·12-s + 0.277i·13-s + 2.27·14-s − 0.793·16-s − 1.73i·17-s − 0.491i·18-s − 1.72·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492691 - 0.116308i\)
\(L(\frac12)\) \(\approx\) \(0.492691 - 0.116308i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 - 2.08iT - 2T^{2} \)
7 \( 1 + 4.08iT - 7T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
17 \( 1 + 7.17iT - 17T^{2} \)
19 \( 1 + 7.52T + 19T^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 + 5.61T + 31T^{2} \)
37 \( 1 + 4.82iT - 37T^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + 2.05iT - 43T^{2} \)
47 \( 1 - 7.49iT - 47T^{2} \)
53 \( 1 + 2.52iT - 53T^{2} \)
59 \( 1 + 4.79T + 59T^{2} \)
61 \( 1 + 7.87T + 61T^{2} \)
67 \( 1 - 6.20iT - 67T^{2} \)
71 \( 1 + 0.475T + 71T^{2} \)
73 \( 1 - 9.35iT - 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 - 16.4iT - 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + 14.8iT - 97T^{2} \)
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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

−9.815846463056146933208481304296, −8.907952001779890389103188456062, −8.060495340149180900989350554348, −7.30374698524120465146301349348, −6.74365968009821856968287480111, −5.69175900421370879491258435638, −4.68490012209616085622061891968, −4.22656384997304596936850672538, −2.64789963582575114938239944238, −0.21802254763783441058039439860, 1.74136608382934803234922070737, 2.39933834608186040508962460746, 3.32458652095562452090158448728, 4.60023138551938391270253571075, 5.74474133157551055222674664931, 6.44518176512432791076049961622, 7.926678119402078861264336794320, 8.587439518791430681615278200643, 9.278896121323871673500608568454, 10.53353637054385676816957157188

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