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Properties

Label 2-1175-5.4-c1-0-7
Degree $2$
Conductor $1175$
Sign $-0.894 - 0.447i$
Analytic cond. $9.38242$
Root an. cond. $3.06307$
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  + 1.49i·2-s − 2.14i·3-s − 0.247·4-s + 3.21·6-s + 0.914i·7-s + 2.62i·8-s − 1.59·9-s − 5.69·11-s + 0.529i·12-s + 5.13i·13-s − 1.37·14-s − 4.43·16-s + 5.54i·17-s − 2.38i·18-s − 4.24·19-s + ⋯
L(s)  = 1  + 1.05i·2-s − 1.23i·3-s − 0.123·4-s + 1.31·6-s + 0.345i·7-s + 0.929i·8-s − 0.530·9-s − 1.71·11-s + 0.152i·12-s + 1.42i·13-s − 0.366·14-s − 1.10·16-s + 1.34i·17-s − 0.562i·18-s − 0.974·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1175\)    =    \(5^{2} \cdot 47\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(9.38242\)
Root analytic conductor: \(3.06307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1175} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1175,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8947483287\)
\(L(\frac12)\) \(\approx\) \(0.8947483287\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 + iT \)
good2 \( 1 - 1.49iT - 2T^{2} \)
3 \( 1 + 2.14iT - 3T^{2} \)
7 \( 1 - 0.914iT - 7T^{2} \)
11 \( 1 + 5.69T + 11T^{2} \)
13 \( 1 - 5.13iT - 13T^{2} \)
17 \( 1 - 5.54iT - 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 7.15iT - 23T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 + 6.52T + 31T^{2} \)
37 \( 1 - 10.0iT - 37T^{2} \)
41 \( 1 + 4.42T + 41T^{2} \)
43 \( 1 + 0.310iT - 43T^{2} \)
53 \( 1 - 6.38iT - 53T^{2} \)
59 \( 1 + 1.42T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 - 4.25iT - 67T^{2} \)
71 \( 1 - 3.83T + 71T^{2} \)
73 \( 1 - 9.86iT - 73T^{2} \)
79 \( 1 - 6.00T + 79T^{2} \)
83 \( 1 + 2.43iT - 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.17773324576703212008311620946, −8.625015168973995132706123642264, −8.378832533187865133382671395587, −7.51863913765978532224363830319, −6.75899636674786404682069389518, −6.27828168074524744317000955215, −5.39484765504620048517742023806, −4.33618671598434252806286185694, −2.49504440988735015446668985542, −1.88295807395996835666640592141, 0.34546840690278793950712476100, 2.23344886613769224862470998448, 3.19567014770483119928143078222, 3.84852937176315755399834027882, 5.04792184022539180024785478620, 5.55146955774374873743314843180, 7.16283628321525296642764659546, 7.79429113199101101151253764660, 9.038207955480830816865371442583, 9.813218303379878618527794478647

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