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Properties

Label 2-1805-5.4-c1-0-75
Degree $2$
Conductor $1805$
Sign $-0.144 - 0.989i$
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  + 1.85i·2-s + 1.45i·3-s − 1.43·4-s + (−0.323 − 2.21i)5-s − 2.69·6-s − 0.477i·7-s + 1.05i·8-s + 0.884·9-s + (4.09 − 0.599i)10-s + 1.95·11-s − 2.08i·12-s − 3.06i·13-s + 0.884·14-s + (3.21 − 0.470i)15-s − 4.81·16-s − 0.973i·17-s + ⋯
L(s)  = 1  + 1.30i·2-s + 0.839i·3-s − 0.715·4-s + (−0.144 − 0.989i)5-s − 1.09·6-s − 0.180i·7-s + 0.372i·8-s + 0.294·9-s + (1.29 − 0.189i)10-s + 0.590·11-s − 0.601i·12-s − 0.848i·13-s + 0.236·14-s + (0.830 − 0.121i)15-s − 1.20·16-s − 0.236i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.144 - 0.989i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.922111567\)
\(L(\frac12)\) \(\approx\) \(1.922111567\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.323 + 2.21i)T \)
19 \( 1 \)
good2 \( 1 - 1.85iT - 2T^{2} \)
3 \( 1 - 1.45iT - 3T^{2} \)
7 \( 1 + 0.477iT - 7T^{2} \)
11 \( 1 - 1.95T + 11T^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
17 \( 1 + 0.973iT - 17T^{2} \)
23 \( 1 + 5.59iT - 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 5.60T + 31T^{2} \)
37 \( 1 + 5.65iT - 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + 2.47iT - 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 + 7.32T + 59T^{2} \)
61 \( 1 - 8.76T + 61T^{2} \)
67 \( 1 - 8.82iT - 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 4.97iT - 73T^{2} \)
79 \( 1 - 0.707T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + 2.14T + 89T^{2} \)
97 \( 1 - 5.89iT - 97T^{2} \)
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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

−9.177823464022109899777611225453, −8.647117379035862577922863013286, −7.917819959191852898896991054672, −7.15243086588086070601565188056, −6.21843123817970572476417085270, −5.50095680889391479815287981700, −4.46427113993832219149017263270, −4.35692375262843395616725477141, −2.75072145634692179094511361388, −0.941820457394683352348054101935, 1.05717180622457846228864775271, 1.99265392155761593335848282461, 2.79467399079224633820581328180, 3.76357855640127859806147096024, 4.54312206636457763645313913291, 6.17459594241892935521014856949, 6.70010998245934217882034265208, 7.34061787857473870544708518254, 8.312289912155168795583837465356, 9.368553630817829220783594785155

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