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Properties

Label 2-1805-5.4-c1-0-10
Degree $2$
Conductor $1805$
Sign $-0.0500 + 0.998i$
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  + 1.47i·2-s + 2.48i·3-s − 0.187·4-s + (0.111 − 2.23i)5-s − 3.67·6-s + 3.24i·7-s + 2.68i·8-s − 3.16·9-s + (3.30 + 0.165i)10-s − 4.18·11-s − 0.466i·12-s + 1.78i·13-s − 4.80·14-s + (5.54 + 0.278i)15-s − 4.34·16-s − 6.33i·17-s + ⋯
L(s)  = 1  + 1.04i·2-s + 1.43i·3-s − 0.0939·4-s + (0.0500 − 0.998i)5-s − 1.49·6-s + 1.22i·7-s + 0.947i·8-s − 1.05·9-s + (1.04 + 0.0523i)10-s − 1.26·11-s − 0.134i·12-s + 0.494i·13-s − 1.28·14-s + (1.43 + 0.0718i)15-s − 1.08·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0500 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0500 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.0500 + 0.998i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.0500 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9803507146\)
\(L(\frac12)\) \(\approx\) \(0.9803507146\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.111 + 2.23i)T \)
19 \( 1 \)
good2 \( 1 - 1.47iT - 2T^{2} \)
3 \( 1 - 2.48iT - 3T^{2} \)
7 \( 1 - 3.24iT - 7T^{2} \)
11 \( 1 + 4.18T + 11T^{2} \)
13 \( 1 - 1.78iT - 13T^{2} \)
17 \( 1 + 6.33iT - 17T^{2} \)
23 \( 1 - 1.43iT - 23T^{2} \)
29 \( 1 - 0.339T + 29T^{2} \)
31 \( 1 + 2.77T + 31T^{2} \)
37 \( 1 - 2.70iT - 37T^{2} \)
41 \( 1 + 7.13T + 41T^{2} \)
43 \( 1 - 9.89iT - 43T^{2} \)
47 \( 1 - 0.445iT - 47T^{2} \)
53 \( 1 + 7.23iT - 53T^{2} \)
59 \( 1 + 3.14T + 59T^{2} \)
61 \( 1 + 3.06T + 61T^{2} \)
67 \( 1 + 8.55iT - 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 1.82iT - 73T^{2} \)
79 \( 1 + 0.698T + 79T^{2} \)
83 \( 1 + 0.552iT - 83T^{2} \)
89 \( 1 + 6.82T + 89T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−9.428519873551114986187534945290, −9.164848801504380608477817521124, −8.311774403016473652421697084399, −7.69446146428180160427030076634, −6.51388965070879102058734019315, −5.48692468638862175740417510613, −5.12690342873288386485577241124, −4.64642673254686390677666004624, −3.17641447323366675744349901680, −2.17748565092820516439596629824, 0.33135595542958049181007032156, 1.58387438575969911085381352056, 2.36063805844045900802543585486, 3.26125092018134709504010754967, 4.08793474607308300479769818848, 5.68953942482013250238554800199, 6.53098613191076499279016561881, 7.20849032196935369843607664555, 7.63555281193957181839466553481, 8.476272295242539364351836692584

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