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Properties

Label 2-1805-5.4-c1-0-8
Degree $2$
Conductor $1805$
Sign $0.824 + 0.565i$
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  − 2.31i·2-s − 2.90i·3-s − 3.35·4-s + (1.84 + 1.26i)5-s − 6.72·6-s + 2.34i·7-s + 3.14i·8-s − 5.43·9-s + (2.92 − 4.26i)10-s − 3.45·11-s + 9.74i·12-s + 3.29i·13-s + 5.43·14-s + (3.67 − 5.35i)15-s + 0.554·16-s + 1.38i·17-s + ⋯
L(s)  = 1  − 1.63i·2-s − 1.67i·3-s − 1.67·4-s + (0.824 + 0.565i)5-s − 2.74·6-s + 0.886i·7-s + 1.11i·8-s − 1.81·9-s + (0.926 − 1.34i)10-s − 1.04·11-s + 2.81i·12-s + 0.912i·13-s + 1.45·14-s + (0.948 − 1.38i)15-s + 0.138·16-s + 0.335i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.824 + 0.565i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.824 + 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6894143023\)
\(L(\frac12)\) \(\approx\) \(0.6894143023\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.84 - 1.26i)T \)
19 \( 1 \)
good2 \( 1 + 2.31iT - 2T^{2} \)
3 \( 1 + 2.90iT - 3T^{2} \)
7 \( 1 - 2.34iT - 7T^{2} \)
11 \( 1 + 3.45T + 11T^{2} \)
13 \( 1 - 3.29iT - 13T^{2} \)
17 \( 1 - 1.38iT - 17T^{2} \)
23 \( 1 - 8.90iT - 23T^{2} \)
29 \( 1 + 6.28T + 29T^{2} \)
31 \( 1 + 7.39T + 31T^{2} \)
37 \( 1 + 0.650iT - 37T^{2} \)
41 \( 1 - 2.01T + 41T^{2} \)
43 \( 1 - 1.72iT - 43T^{2} \)
47 \( 1 + 6.32iT - 47T^{2} \)
53 \( 1 + 2.12iT - 53T^{2} \)
59 \( 1 - 6.06T + 59T^{2} \)
61 \( 1 - 2.96T + 61T^{2} \)
67 \( 1 - 2.46iT - 67T^{2} \)
71 \( 1 + 5.69T + 71T^{2} \)
73 \( 1 + 6.88iT - 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 - 1.84iT - 83T^{2} \)
89 \( 1 + 6.47T + 89T^{2} \)
97 \( 1 - 7.61iT - 97T^{2} \)
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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

−9.298037334642535395057334274865, −8.679352121527169787045792165558, −7.56102361834047121081196218070, −6.94130319180305493677597946789, −5.81536636175508035661149122761, −5.35613227233981643088585931540, −3.65493224761963499935815298736, −2.68600582600942100953694334951, −2.02680421317413311491091149655, −1.54300290347935757630190411638, 0.24019316393194579885862825516, 2.70397196583524763299025160829, 4.04826157389403517254781114657, 4.67090305905614074448002690306, 5.43817832104994140763245183398, 5.76650058829523922105554630377, 6.95238698690841278356003312287, 7.83979308720155290421652842189, 8.580299254901305534144602040009, 9.193624321263350486978877220808

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