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Properties

Label 2-1805-5.4-c1-0-144
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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.406i·2-s − 3.27i·3-s + 1.83·4-s + 2.23·5-s − 1.33·6-s − 1.55i·8-s − 7.74·9-s − 0.907i·10-s − 2.92·11-s − 6.01i·12-s − 6.51i·13-s − 7.32i·15-s + 3.03·16-s + 3.14i·18-s + 4.10·20-s + ⋯
L(s)  = 1  − 0.287i·2-s − 1.89i·3-s + 0.917·4-s + 0.999·5-s − 0.543·6-s − 0.550i·8-s − 2.58·9-s − 0.287i·10-s − 0.883·11-s − 1.73i·12-s − 1.80i·13-s − 1.89i·15-s + 0.759·16-s + 0.740i·18-s + 0.917·20-s + ⋯

Functional equation

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

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: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.235091960\)
\(L(\frac12)\) \(\approx\) \(2.235091960\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23T \)
19 \( 1 \)
good2 \( 1 + 0.406iT - 2T^{2} \)
3 \( 1 + 3.27iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 2.92T + 11T^{2} \)
13 \( 1 + 6.51iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.01iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 5.09iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 1.11T + 61T^{2} \)
67 \( 1 - 8.95iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−8.509550134992351709009694992092, −7.961896732253564231752869627915, −7.24042970097347857682234561824, −6.57041741355175367639719538577, −5.77095967158351309633919280830, −5.36475523377751444023215872102, −3.04397293363790477363660306673, −2.65522743083767472106731319547, −1.71619025018028218358944602134, −0.75997196959084097768894681056, 2.05666627198079070150501027710, 2.82854609300446008671748082859, 3.95711360073062450895337654813, 4.86470323083272323222422149044, 5.58263341261129248233136420625, 6.22749152864780458247941444768, 7.17352131539734235670374723803, 8.330121478703706756448146179072, 9.175644064211422021397974889799, 9.578834201477413694845550589891

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