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Properties

Label 2-1805-5.4-c1-0-106
Degree $2$
Conductor $1805$
Sign $0.145 + 0.989i$
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  − 1.61i·2-s + 1.18i·3-s − 0.621·4-s + (0.326 + 2.21i)5-s + 1.92·6-s − 2.23i·7-s − 2.23i·8-s + 1.58·9-s + (3.58 − 0.528i)10-s + 5.64·11-s − 0.738i·12-s − 4.70i·13-s − 3.61·14-s + (−2.62 + 0.387i)15-s − 4.85·16-s + 0.785i·17-s + ⋯
L(s)  = 1  − 1.14i·2-s + 0.686i·3-s − 0.310·4-s + (0.145 + 0.989i)5-s + 0.785·6-s − 0.843i·7-s − 0.789i·8-s + 0.529·9-s + (1.13 − 0.167i)10-s + 1.70·11-s − 0.213i·12-s − 1.30i·13-s − 0.965·14-s + (−0.678 + 0.100i)15-s − 1.21·16-s + 0.190i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.145 + 0.989i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.145 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.209026421\)
\(L(\frac12)\) \(\approx\) \(2.209026421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.326 - 2.21i)T \)
19 \( 1 \)
good2 \( 1 + 1.61iT - 2T^{2} \)
3 \( 1 - 1.18iT - 3T^{2} \)
7 \( 1 + 2.23iT - 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 4.70iT - 13T^{2} \)
17 \( 1 - 0.785iT - 17T^{2} \)
23 \( 1 + 5.13iT - 23T^{2} \)
29 \( 1 - 3.03T + 29T^{2} \)
31 \( 1 + 8.10T + 31T^{2} \)
37 \( 1 + 0.985iT - 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 1.52iT - 43T^{2} \)
47 \( 1 + 0.960iT - 47T^{2} \)
53 \( 1 + 4.41iT - 53T^{2} \)
59 \( 1 - 9.87T + 59T^{2} \)
61 \( 1 - 2.09T + 61T^{2} \)
67 \( 1 + 3.24iT - 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 + 7.52iT - 83T^{2} \)
89 \( 1 - 3.40T + 89T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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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

−9.565476658675267202135965746353, −8.596031622074731535839763336470, −7.22687861661002992016557224825, −6.90061767827747040569283454508, −5.91068193194181535575677922846, −4.50504148284350918796681792445, −3.74005730774920802398769927850, −3.34050683006761076606713362194, −2.07461849701251986860432287556, −0.907486348999468447927066288189, 1.41178480266083375573881974068, 2.07517567576090562923489374331, 3.90943950861504640513360626004, 4.76238847520415741152951855749, 5.70194667756026619267284698659, 6.36709524660493635457733532638, 7.00212758368743773986373559872, 7.67973942613501926836433079737, 8.726450499529003982413449681220, 9.051264208838829479830652464270

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