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Properties

Label 2-1805-95.59-c0-0-2
Degree $2$
Conductor $1805$
Sign $-0.934 - 0.356i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.245 − 1.39i)2-s + (−1.08 − 0.909i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−1.53 + 1.28i)6-s + (0.173 + 0.984i)9-s + (−0.245 − 1.39i)10-s + (0.707 + 1.22i)12-s + (1.08 − 0.909i)13-s + (−1.32 − 0.483i)15-s + (−0.766 − 0.642i)16-s + 1.41·18-s − 1.00·20-s + (0.766 − 0.642i)25-s + (−1 − 1.73i)26-s + ⋯
L(s)  = 1  + (0.245 − 1.39i)2-s + (−1.08 − 0.909i)3-s + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)5-s + (−1.53 + 1.28i)6-s + (0.173 + 0.984i)9-s + (−0.245 − 1.39i)10-s + (0.707 + 1.22i)12-s + (1.08 − 0.909i)13-s + (−1.32 − 0.483i)15-s + (−0.766 − 0.642i)16-s + 1.41·18-s − 1.00·20-s + (0.766 − 0.642i)25-s + (−1 − 1.73i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.934 - 0.356i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.934 - 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.085406930\)
\(L(\frac12)\) \(\approx\) \(1.085406930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 \)
good2 \( 1 + (-0.245 + 1.39i)T + (-0.939 - 0.342i)T^{2} \)
3 \( 1 + (1.08 + 0.909i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.08 + 0.909i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.262231261466289339912597498311, −8.421998264284717149198309315362, −7.25118437079749425514440134685, −6.43933085251581230731965191783, −5.68775212760742020403063258391, −5.04768519434017481305142148611, −3.83643710734774335582683802024, −2.73519919670350252816794925419, −1.65094833084799794362201208586, −0.936159941224155285752364785320, 1.89668163392405352790239876121, 3.62810874121219970434365562631, 4.58214825888739054765716131936, 5.32106214221683999509938588298, 5.88353821376375533656595006420, 6.53847736812681865164033266198, 7.07720885032597282744008589829, 8.345697732356436701351881240427, 9.041540071319669330068615292381, 9.863959226738636718264688634445

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