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Properties

Label 2-1805-5.4-c1-0-46
Degree $2$
Conductor $1805$
Sign $-0.982 + 0.186i$
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  + 1.74i·2-s + 0.766i·3-s − 1.03·4-s + (2.19 − 0.415i)5-s − 1.33·6-s + 2.32i·7-s + 1.68i·8-s + 2.41·9-s + (0.724 + 3.82i)10-s − 4.04·11-s − 0.791i·12-s + 3.88i·13-s − 4.05·14-s + (0.318 + 1.68i)15-s − 4.99·16-s + 4.16i·17-s + ⋯
L(s)  = 1  + 1.23i·2-s + 0.442i·3-s − 0.516·4-s + (0.982 − 0.186i)5-s − 0.544·6-s + 0.879i·7-s + 0.595i·8-s + 0.804·9-s + (0.229 + 1.20i)10-s − 1.21·11-s − 0.228i·12-s + 1.07i·13-s − 1.08·14-s + (0.0823 + 0.434i)15-s − 1.24·16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.982 + 0.186i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.982 + 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.042103520\)
\(L(\frac12)\) \(\approx\) \(2.042103520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.19 + 0.415i)T \)
19 \( 1 \)
good2 \( 1 - 1.74iT - 2T^{2} \)
3 \( 1 - 0.766iT - 3T^{2} \)
7 \( 1 - 2.32iT - 7T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 - 3.88iT - 13T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
23 \( 1 + 7.90iT - 23T^{2} \)
29 \( 1 - 7.12T + 29T^{2} \)
31 \( 1 + 2.72T + 31T^{2} \)
37 \( 1 - 3.70iT - 37T^{2} \)
41 \( 1 + 3.50T + 41T^{2} \)
43 \( 1 + 9.37iT - 43T^{2} \)
47 \( 1 + 0.333iT - 47T^{2} \)
53 \( 1 - 6.90iT - 53T^{2} \)
59 \( 1 + 7.15T + 59T^{2} \)
61 \( 1 + 1.35T + 61T^{2} \)
67 \( 1 - 7.79iT - 67T^{2} \)
71 \( 1 + 0.0586T + 71T^{2} \)
73 \( 1 + 9.15iT - 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 3.84iT - 97T^{2} \)
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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

−9.470605830376755564291748725857, −8.687505413868794770882364126888, −8.270646983651417565729880003895, −7.09156957291702146455550799565, −6.47297067841980230929883046680, −5.76760593603131218024651464011, −5.00691026778916735674801602020, −4.38162445303763043993232949657, −2.66617339653764752229284300653, −1.86583933209171501403104370821, 0.73798066306842341780953204097, 1.67166698790392099790704150950, 2.69194688441612060761652598475, 3.37884856926269873815708945019, 4.63770534668172102582030075416, 5.46468741807191696292663919595, 6.58412640328035232372788087502, 7.32654502029759502470664833595, 7.913802445444560004760788227172, 9.327630118813928957468459860867

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