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Properties

Label 2-1805-5.4-c1-0-116
Degree $2$
Conductor $1805$
Sign $-0.525 - 0.851i$
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  − 2.37i·2-s − 2.28i·3-s − 3.62·4-s + (1.17 + 1.90i)5-s − 5.41·6-s + 1.63i·7-s + 3.84i·8-s − 2.22·9-s + (4.51 − 2.78i)10-s + 2.72·11-s + 8.27i·12-s − 6.19i·13-s + 3.88·14-s + (4.34 − 2.68i)15-s + 1.87·16-s − 3.12i·17-s + ⋯
L(s)  = 1  − 1.67i·2-s − 1.31i·3-s − 1.81·4-s + (0.525 + 0.851i)5-s − 2.21·6-s + 0.618i·7-s + 1.35i·8-s − 0.740·9-s + (1.42 − 0.880i)10-s + 0.822·11-s + 2.38i·12-s − 1.71i·13-s + 1.03·14-s + (1.12 − 0.692i)15-s + 0.468·16-s − 0.757i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.525 - 0.851i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.525 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.557654786\)
\(L(\frac12)\) \(\approx\) \(1.557654786\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.17 - 1.90i)T \)
19 \( 1 \)
good2 \( 1 + 2.37iT - 2T^{2} \)
3 \( 1 + 2.28iT - 3T^{2} \)
7 \( 1 - 1.63iT - 7T^{2} \)
11 \( 1 - 2.72T + 11T^{2} \)
13 \( 1 + 6.19iT - 13T^{2} \)
17 \( 1 + 3.12iT - 17T^{2} \)
23 \( 1 + 7.29iT - 23T^{2} \)
29 \( 1 + 2.22T + 29T^{2} \)
31 \( 1 - 4.42T + 31T^{2} \)
37 \( 1 + 2.04iT - 37T^{2} \)
41 \( 1 - 3.92T + 41T^{2} \)
43 \( 1 + 0.472iT - 43T^{2} \)
47 \( 1 - 2.30iT - 47T^{2} \)
53 \( 1 - 6.36iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 4.79T + 61T^{2} \)
67 \( 1 + 0.670iT - 67T^{2} \)
71 \( 1 - 2.63T + 71T^{2} \)
73 \( 1 - 6.51iT - 73T^{2} \)
79 \( 1 + 4.12T + 79T^{2} \)
83 \( 1 + 6.42iT - 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + 0.129iT - 97T^{2} \)
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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

−8.964484925688084260033672250358, −8.081294654802260536903805701969, −7.18342577308579406431182476024, −6.32925145226341847425727040863, −5.58032139375981697240820182446, −4.30078080443082009305145428291, −2.90499087162014245604724269565, −2.70131851182218418119750261079, −1.64047561474941804153419975700, −0.60831625387489208048037515832, 1.56412054326970490897306360012, 3.81234497394322537679292470467, 4.29217197426546207588781062594, 4.92315313399621477929746050116, 5.80804294077408709687613736037, 6.50889949817815173173362591058, 7.33085015516509272388625493995, 8.304696429670910647785178586888, 9.142968561274235811161360783866, 9.354247392316904937158708103396

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