Properties

Label 2-1805-5.4-c1-0-12
Degree $2$
Conductor $1805$
Sign $-0.0551 + 0.998i$
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  + 2.42i·2-s + 1.05i·3-s − 3.89·4-s + (0.123 − 2.23i)5-s − 2.56·6-s + 4.52i·7-s − 4.59i·8-s + 1.88·9-s + (5.41 + 0.299i)10-s + 0.194·11-s − 4.11i·12-s − 3.48i·13-s − 10.9·14-s + (2.35 + 0.130i)15-s + 3.36·16-s + 6.60i·17-s + ⋯
L(s)  = 1  + 1.71i·2-s + 0.610i·3-s − 1.94·4-s + (0.0551 − 0.998i)5-s − 1.04·6-s + 1.71i·7-s − 1.62i·8-s + 0.627·9-s + (1.71 + 0.0946i)10-s + 0.0586·11-s − 1.18i·12-s − 0.967i·13-s − 2.93·14-s + (0.609 + 0.0336i)15-s + 0.840·16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0551 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0551 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.0551 + 0.998i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.0551 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.009361904\)
\(L(\frac12)\) \(\approx\) \(1.009361904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.123 + 2.23i)T \)
19 \( 1 \)
good2 \( 1 - 2.42iT - 2T^{2} \)
3 \( 1 - 1.05iT - 3T^{2} \)
7 \( 1 - 4.52iT - 7T^{2} \)
11 \( 1 - 0.194T + 11T^{2} \)
13 \( 1 + 3.48iT - 13T^{2} \)
17 \( 1 - 6.60iT - 17T^{2} \)
23 \( 1 - 1.18iT - 23T^{2} \)
29 \( 1 + 4.10T + 29T^{2} \)
31 \( 1 + 5.06T + 31T^{2} \)
37 \( 1 - 4.20iT - 37T^{2} \)
41 \( 1 + 9.20T + 41T^{2} \)
43 \( 1 - 2.26iT - 43T^{2} \)
47 \( 1 - 5.95iT - 47T^{2} \)
53 \( 1 + 6.94iT - 53T^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 - 4.79T + 61T^{2} \)
67 \( 1 - 5.22iT - 67T^{2} \)
71 \( 1 + 5.85T + 71T^{2} \)
73 \( 1 - 6.54iT - 73T^{2} \)
79 \( 1 - 5.47T + 79T^{2} \)
83 \( 1 + 9.54iT - 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 7.41iT - 97T^{2} \)
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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

−9.495848699797589934042903810870, −8.744173892313296055282188224795, −8.383341576194444212276197400167, −7.64015589288090585805440143331, −6.48649134714761267855778537459, −5.65083962188155322403972502418, −5.36053618292661975721715378849, −4.52382786196445243695365723694, −3.53851442172795864971454798987, −1.77046822498661603395191794661, 0.38035134702239371389049874510, 1.53768971657821667013214081041, 2.37821466315359800371746047657, 3.56159129239665729455184371246, 4.03053664231171720689987245727, 4.99591971708064643746111507678, 6.67710701324565617354988495165, 7.09162203973035798510947976704, 7.74471226114101440959801040383, 9.172152369865921250768539332130

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