Properties

Label 2-1805-1.1-c1-0-94
Degree $2$
Conductor $1805$
Sign $-1$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.43·2-s − 2.94·3-s + 3.94·4-s + 5-s − 7.18·6-s − 3.82·7-s + 4.74·8-s + 5.67·9-s + 2.43·10-s − 2.12·11-s − 11.6·12-s + 3.65·13-s − 9.31·14-s − 2.94·15-s + 3.67·16-s − 3.04·17-s + 13.8·18-s + 3.94·20-s + 11.2·21-s − 5.18·22-s − 4.81·23-s − 13.9·24-s + 25-s + 8.91·26-s − 7.87·27-s − 15.0·28-s − 6.03·29-s + ⋯
L(s)  = 1  + 1.72·2-s − 1.70·3-s + 1.97·4-s + 0.447·5-s − 2.93·6-s − 1.44·7-s + 1.67·8-s + 1.89·9-s + 0.771·10-s − 0.640·11-s − 3.35·12-s + 1.01·13-s − 2.48·14-s − 0.760·15-s + 0.918·16-s − 0.737·17-s + 3.26·18-s + 0.882·20-s + 2.45·21-s − 1.10·22-s − 1.00·23-s − 2.85·24-s + 0.200·25-s + 1.74·26-s − 1.51·27-s − 2.84·28-s − 1.12·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
19 \( 1 \)
good2 \( 1 - 2.43T + 2T^{2} \)
3 \( 1 + 2.94T + 3T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 + 2.12T + 11T^{2} \)
13 \( 1 - 3.65T + 13T^{2} \)
17 \( 1 + 3.04T + 17T^{2} \)
23 \( 1 + 4.81T + 23T^{2} \)
29 \( 1 + 6.03T + 29T^{2} \)
31 \( 1 + 3.57T + 31T^{2} \)
37 \( 1 + 3.93T + 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 + 5.60T + 43T^{2} \)
47 \( 1 + 8.41T + 47T^{2} \)
53 \( 1 + 4.80T + 53T^{2} \)
59 \( 1 - 5.13T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 5.38T + 67T^{2} \)
71 \( 1 - 0.123T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 4.25T + 79T^{2} \)
83 \( 1 - 4.39T + 83T^{2} \)
89 \( 1 - 0.0772T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−9.114036470987235002310880227193, −7.50522270546496593451649728010, −6.56616277404430520351095485647, −6.22945408323817731560892786735, −5.69927217887504531427148417646, −4.96598275610311573587609703319, −4.03134722650489596858673180680, −3.23107238742905553925382255043, −1.89507769897254834967821868741, 0, 1.89507769897254834967821868741, 3.23107238742905553925382255043, 4.03134722650489596858673180680, 4.96598275610311573587609703319, 5.69927217887504531427148417646, 6.22945408323817731560892786735, 6.56616277404430520351095485647, 7.50522270546496593451649728010, 9.114036470987235002310880227193

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