Properties

Label 2-1805-1.1-c3-0-118
Degree $2$
Conductor $1805$
Sign $-1$
Analytic cond. $106.498$
Root an. cond. $10.3198$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 8·4-s − 5·5-s − 22·7-s − 11·9-s − 12·11-s + 32·12-s − 8·13-s + 20·15-s + 64·16-s − 66·17-s + 40·20-s + 88·21-s − 30·23-s + 25·25-s + 152·27-s + 176·28-s + 6·29-s + 64·31-s + 48·33-s + 110·35-s + 88·36-s + 16·37-s + 32·39-s − 54·41-s + 182·43-s + 96·44-s + ⋯
L(s)  = 1  − 0.769·3-s − 4-s − 0.447·5-s − 1.18·7-s − 0.407·9-s − 0.328·11-s + 0.769·12-s − 0.170·13-s + 0.344·15-s + 16-s − 0.941·17-s + 0.447·20-s + 0.914·21-s − 0.271·23-s + 1/5·25-s + 1.08·27-s + 1.18·28-s + 0.0384·29-s + 0.370·31-s + 0.253·33-s + 0.531·35-s + 0.407·36-s + 0.0710·37-s + 0.131·39-s − 0.205·41-s + 0.645·43-s + 0.328·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(106.498\)
Root analytic conductor: \(10.3198\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1805,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p T \)
19 \( 1 \)
good2 \( 1 + p^{3} T^{2} \)
3 \( 1 + 4 T + p^{3} T^{2} \)
7 \( 1 + 22 T + p^{3} T^{2} \)
11 \( 1 + 12 T + p^{3} T^{2} \)
13 \( 1 + 8 T + p^{3} T^{2} \)
17 \( 1 + 66 T + p^{3} T^{2} \)
23 \( 1 + 30 T + p^{3} T^{2} \)
29 \( 1 - 6 T + p^{3} T^{2} \)
31 \( 1 - 64 T + p^{3} T^{2} \)
37 \( 1 - 16 T + p^{3} T^{2} \)
41 \( 1 + 54 T + p^{3} T^{2} \)
43 \( 1 - 182 T + p^{3} T^{2} \)
47 \( 1 - 594 T + p^{3} T^{2} \)
53 \( 1 + 396 T + p^{3} T^{2} \)
59 \( 1 - 564 T + p^{3} T^{2} \)
61 \( 1 + 706 T + p^{3} T^{2} \)
67 \( 1 - 628 T + p^{3} T^{2} \)
71 \( 1 - 984 T + p^{3} T^{2} \)
73 \( 1 - 14 T + p^{3} T^{2} \)
79 \( 1 - 328 T + p^{3} T^{2} \)
83 \( 1 + 294 T + p^{3} T^{2} \)
89 \( 1 + 918 T + p^{3} T^{2} \)
97 \( 1 - 1564 T + p^{3} T^{2} \)
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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

−8.643085153842114377279759444911, −7.79221328529797416176557759019, −6.76022256239551205413868706321, −6.05690168919532583195305379024, −5.26805310904802031185229077182, −4.43641129561021390388854046984, −3.58174434307625376212799229353, −2.59208316294082012977778571460, −0.70795389237213369226487459304, 0, 0.70795389237213369226487459304, 2.59208316294082012977778571460, 3.58174434307625376212799229353, 4.43641129561021390388854046984, 5.26805310904802031185229077182, 6.05690168919532583195305379024, 6.76022256239551205413868706321, 7.79221328529797416176557759019, 8.643085153842114377279759444911

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