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Properties

Label 2-1805-5.4-c1-0-25
Degree $2$
Conductor $1805$
Sign $-0.165 + 0.986i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  + 0.906i·2-s + 3.21i·3-s + 1.17·4-s + (−0.370 + 2.20i)5-s − 2.91·6-s + 2.59i·7-s + 2.88i·8-s − 7.35·9-s + (−1.99 − 0.336i)10-s + 0.741·11-s + 3.78i·12-s − 3.78i·13-s − 2.35·14-s + (−7.09 − 1.19i)15-s − 0.258·16-s − 3.16i·17-s + ⋯
L(s)  = 1  + 0.641i·2-s + 1.85i·3-s + 0.588·4-s + (−0.165 + 0.986i)5-s − 1.19·6-s + 0.981i·7-s + 1.01i·8-s − 2.45·9-s + (−0.632 − 0.106i)10-s + 0.223·11-s + 1.09i·12-s − 1.05i·13-s − 0.629·14-s + (−1.83 − 0.307i)15-s − 0.0647·16-s − 0.768i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.165 + 0.986i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.617309388\)
\(L(\frac12)\) \(\approx\) \(1.617309388\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.370 - 2.20i)T \)
19 \( 1 \)
good2 \( 1 - 0.906iT - 2T^{2} \)
3 \( 1 - 3.21iT - 3T^{2} \)
7 \( 1 - 2.59iT - 7T^{2} \)
11 \( 1 - 0.741T + 11T^{2} \)
13 \( 1 + 3.78iT - 13T^{2} \)
17 \( 1 + 3.16iT - 17T^{2} \)
23 \( 1 + 0.570iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 - 1.40iT - 37T^{2} \)
41 \( 1 - 3.83T + 41T^{2} \)
43 \( 1 - 2.59iT - 43T^{2} \)
47 \( 1 - 5.08iT - 47T^{2} \)
53 \( 1 + 0.160iT - 53T^{2} \)
59 \( 1 - 8.35T + 59T^{2} \)
61 \( 1 + 8.57T + 61T^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 1.83T + 79T^{2} \)
83 \( 1 - 4.19iT - 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 3.78iT - 97T^{2} \)
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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

−9.955266284752761071810315092736, −9.039417160184400738884114935614, −8.362540024540271699773548842380, −7.51358612360136525161004575910, −6.46755163571395679876294546876, −5.68651354198141334621635589544, −5.21811775788673945244287965001, −4.08474769651222406474469935681, −2.93365441519194347555078878973, −2.65973443003436196141664137675, 0.57597174279959221466645074830, 1.49510889087066935215285968382, 2.05448114651725341620486809027, 3.40586582223067346002101311999, 4.36095544569600037484058357968, 5.74832841398024574783478767317, 6.52833392468048732638498094844, 7.15065300286638672405327198868, 7.73816752976408927980021209705, 8.572833211721123488298494348695

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