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Properties

Label 2-1805-5.4-c1-0-31
Degree $2$
Conductor $1805$
Sign $0.855 - 0.518i$
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  + 0.862i·2-s − 3.07i·3-s + 1.25·4-s + (−1.91 + 1.15i)5-s + 2.65·6-s + 0.566i·7-s + 2.80i·8-s − 6.48·9-s + (−1 − 1.64i)10-s − 1.91·11-s − 3.86i·12-s − 0.194i·13-s − 0.488·14-s + (3.56 + 5.88i)15-s + 0.0877·16-s + 5.29i·17-s + ⋯
L(s)  = 1  + 0.610i·2-s − 1.77i·3-s + 0.627·4-s + (−0.855 + 0.518i)5-s + 1.08·6-s + 0.214i·7-s + 0.993i·8-s − 2.16·9-s + (−0.316 − 0.521i)10-s − 0.576·11-s − 1.11i·12-s − 0.0539i·13-s − 0.130·14-s + (0.921 + 1.52i)15-s + 0.0219·16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.855 - 0.518i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.855 - 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513973730\)
\(L(\frac12)\) \(\approx\) \(1.513973730\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.91 - 1.15i)T \)
19 \( 1 \)
good2 \( 1 - 0.862iT - 2T^{2} \)
3 \( 1 + 3.07iT - 3T^{2} \)
7 \( 1 - 0.566iT - 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
13 \( 1 + 0.194iT - 13T^{2} \)
17 \( 1 - 5.29iT - 17T^{2} \)
23 \( 1 + 3.37iT - 23T^{2} \)
29 \( 1 - 8.73T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 0.955iT - 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 4.93iT - 43T^{2} \)
47 \( 1 - 8.83iT - 47T^{2} \)
53 \( 1 - 8.20iT - 53T^{2} \)
59 \( 1 + 3.71T + 59T^{2} \)
61 \( 1 + 3.51T + 61T^{2} \)
67 \( 1 - 4.04iT - 67T^{2} \)
71 \( 1 - 5.19T + 71T^{2} \)
73 \( 1 + 8.60iT - 73T^{2} \)
79 \( 1 + 6.62T + 79T^{2} \)
83 \( 1 - 4.51iT - 83T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−8.747198109184253167724735471572, −8.023248355093264803054097931806, −7.81730939161368811821654217443, −6.97480944235869535320084588716, −6.29280456243008175595704199821, −5.93240195506646371394961324498, −4.52942162536712509200174289071, −2.91102862025525828316609228866, −2.45573245177319008314493769308, −1.11072972647342762775173189881, 0.63642656528170702943041939064, 2.60611378352894736872901410894, 3.32493686466634782664228709364, 4.15239412261466882574791839022, 4.82925627757407190155363858853, 5.62886906189483106065763545083, 6.89621600726412654814228562828, 7.78704027153460763687444911604, 8.662924865314755441021581394962, 9.433416897827452883594968314019

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