Properties

Label 2-1805-1.1-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.210·2-s + 0.0798·3-s − 1.95·4-s + 5-s + 0.0168·6-s − 1.68·7-s − 0.834·8-s − 2.99·9-s + 0.210·10-s − 2.88·11-s − 0.156·12-s − 6.46·13-s − 0.355·14-s + 0.0798·15-s + 3.73·16-s + 2.98·17-s − 0.631·18-s − 1.95·20-s − 0.134·21-s − 0.609·22-s + 8.25·23-s − 0.0666·24-s + 25-s − 1.36·26-s − 0.478·27-s + 3.29·28-s + 7.25·29-s + ⋯
L(s)  = 1  + 0.149·2-s + 0.0461·3-s − 0.977·4-s + 0.447·5-s + 0.00688·6-s − 0.637·7-s − 0.295·8-s − 0.997·9-s + 0.0667·10-s − 0.870·11-s − 0.0451·12-s − 1.79·13-s − 0.0951·14-s + 0.0206·15-s + 0.933·16-s + 0.724·17-s − 0.148·18-s − 0.437·20-s − 0.0294·21-s − 0.129·22-s + 1.72·23-s − 0.0136·24-s + 0.200·25-s − 0.267·26-s − 0.0921·27-s + 0.623·28-s + 1.34·29-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: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.004196885\)
\(L(\frac12)\) \(\approx\) \(1.004196885\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.210T + 2T^{2} \)
3 \( 1 - 0.0798T + 3T^{2} \)
7 \( 1 + 1.68T + 7T^{2} \)
11 \( 1 + 2.88T + 11T^{2} \)
13 \( 1 + 6.46T + 13T^{2} \)
17 \( 1 - 2.98T + 17T^{2} \)
23 \( 1 - 8.25T + 23T^{2} \)
29 \( 1 - 7.25T + 29T^{2} \)
31 \( 1 - 4.05T + 31T^{2} \)
37 \( 1 - 7.96T + 37T^{2} \)
41 \( 1 - 5.45T + 41T^{2} \)
43 \( 1 + 5.33T + 43T^{2} \)
47 \( 1 - 1.64T + 47T^{2} \)
53 \( 1 - 1.91T + 53T^{2} \)
59 \( 1 - 3.55T + 59T^{2} \)
61 \( 1 + 7.90T + 61T^{2} \)
67 \( 1 - 2.26T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 2.41T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + 9.86T + 97T^{2} \)
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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

−9.458679004941150625528326435211, −8.513725366346648873822519945604, −7.84604125602370095686055924365, −6.87006376106776975786249287208, −5.83867171268605717404431711804, −5.15395942301077313112451787625, −4.57126698094564980259052555794, −3.03873229691883182306050572449, −2.70999092260040075046379288512, −0.64525918014165665122076447559, 0.64525918014165665122076447559, 2.70999092260040075046379288512, 3.03873229691883182306050572449, 4.57126698094564980259052555794, 5.15395942301077313112451787625, 5.83867171268605717404431711804, 6.87006376106776975786249287208, 7.84604125602370095686055924365, 8.513725366346648873822519945604, 9.458679004941150625528326435211

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