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Properties

Label 2-1805-5.4-c1-0-20
Degree $2$
Conductor $1805$
Sign $0.521 - 0.853i$
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  − 1.78i·2-s + 2.38i·3-s − 1.18·4-s + (−1.16 + 1.90i)5-s + 4.25·6-s − 4.23i·7-s − 1.45i·8-s − 2.68·9-s + (3.40 + 2.07i)10-s − 0.490·11-s − 2.82i·12-s + 4.16i·13-s − 7.56·14-s + (−4.54 − 2.77i)15-s − 4.96·16-s + 2.03i·17-s + ⋯
L(s)  = 1  − 1.26i·2-s + 1.37i·3-s − 0.592·4-s + (−0.521 + 0.853i)5-s + 1.73·6-s − 1.60i·7-s − 0.514i·8-s − 0.894·9-s + (1.07 + 0.657i)10-s − 0.148·11-s − 0.815i·12-s + 1.15i·13-s − 2.02·14-s + (−1.17 − 0.717i)15-s − 1.24·16-s + 0.493i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.521 - 0.853i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.181617123\)
\(L(\frac12)\) \(\approx\) \(1.181617123\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.16 - 1.90i)T \)
19 \( 1 \)
good2 \( 1 + 1.78iT - 2T^{2} \)
3 \( 1 - 2.38iT - 3T^{2} \)
7 \( 1 + 4.23iT - 7T^{2} \)
11 \( 1 + 0.490T + 11T^{2} \)
13 \( 1 - 4.16iT - 13T^{2} \)
17 \( 1 - 2.03iT - 17T^{2} \)
23 \( 1 - 4.39iT - 23T^{2} \)
29 \( 1 - 3.26T + 29T^{2} \)
31 \( 1 - 4.08T + 31T^{2} \)
37 \( 1 - 2.14iT - 37T^{2} \)
41 \( 1 - 4.36T + 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + 2.62iT - 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 0.542T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 7.15iT - 67T^{2} \)
71 \( 1 - 6.03T + 71T^{2} \)
73 \( 1 - 2.05iT - 73T^{2} \)
79 \( 1 + 5.34T + 79T^{2} \)
83 \( 1 - 8.11iT - 83T^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 + 5.64iT - 97T^{2} \)
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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

−9.844828615312585292338560477382, −9.100202747150024471954620586560, −7.84508238624256387119091035404, −7.06192804753254888762678674154, −6.28245948855614575252399746794, −4.58094029468374814422374044331, −4.20930958079651123551553689195, −3.58067152346775011854571057129, −2.80035874562101557780311342498, −1.29569575742681786708776328838, 0.45968833278746869349084492853, 2.02033499495490713065147813208, 2.89086651360334270483429188892, 4.69570149645450858890191341042, 5.44516359305814515254672367787, 6.00893217603930526056719933669, 6.78924410055352252016914876257, 7.64093525160708958487381201650, 8.211590053593891073312066217829, 8.587068201607580460770227646314

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