Properties

Label 2-1805-5.4-c1-0-9
Degree $2$
Conductor $1805$
Sign $0.447 - 0.894i$
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  − 2i·2-s − 2·4-s + (−1 + 2i)5-s + 4i·7-s + 3·9-s + (4 + 2i)10-s − 11-s − 2i·13-s + 8·14-s − 4·16-s + 2i·17-s − 6i·18-s + (2 − 4i)20-s + 2i·22-s + 6i·23-s + ⋯
L(s)  = 1  − 1.41i·2-s − 4-s + (−0.447 + 0.894i)5-s + 1.51i·7-s + 9-s + (1.26 + 0.632i)10-s − 0.301·11-s − 0.554i·13-s + 2.13·14-s − 16-s + 0.485i·17-s − 1.41i·18-s + (0.447 − 0.894i)20-s + 0.426i·22-s + 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8128619440\)
\(L(\frac12)\) \(\approx\) \(0.8128619440\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 - 2i)T \)
19 \( 1 \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 - T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−9.517336983817381026615188574883, −9.043107785801728578803226425845, −7.83152404885177440445066360317, −7.20610153750721872675018328112, −6.07954418908341089131109404540, −5.26060202433194636492583668451, −3.97707571169612693924667495769, −3.36108865552823585813803934330, −2.42031440845551366904758511332, −1.66364211856435982528954261813, 0.29250181756487365504488363043, 1.72502237699623100400773022357, 3.71066866067635643536738663027, 4.48592941136036888675538488598, 4.94578460729255212020682293118, 6.09830437590725519054301895606, 7.00590706481127108489076837400, 7.48716003058162486652418858493, 7.942953648120923239406336837282, 9.017668135986333554840430619668

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