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Properties

Label 2-1805-5.4-c1-0-90
Degree $2$
Conductor $1805$
Sign $1$
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  + 1.69i·2-s + 1.52i·3-s − 0.863·4-s − 2.23·5-s − 2.57·6-s + 1.92i·8-s + 0.678·9-s − 3.78i·10-s − 5.95·11-s − 1.31i·12-s − 6.79i·13-s − 3.40i·15-s − 4.98·16-s + 1.14i·18-s + 1.93·20-s + ⋯
L(s)  = 1  + 1.19i·2-s + 0.879i·3-s − 0.431·4-s − 0.999·5-s − 1.05·6-s + 0.679i·8-s + 0.226·9-s − 1.19i·10-s − 1.79·11-s − 0.379i·12-s − 1.88i·13-s − 0.879i·15-s − 1.24·16-s + 0.270i·18-s + 0.431·20-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.5122729325\)
\(L(\frac12)\) \(\approx\) \(0.5122729325\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2.23T \)
19 \( 1 \)
good2 \( 1 - 1.69iT - 2T^{2} \)
3 \( 1 - 1.52iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 5.95T + 11T^{2} \)
13 \( 1 + 6.79iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12.1iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6.04iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 + 3.36iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2.99iT - 97T^{2} \)
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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

−8.993918588116961690521860777594, −8.195327814615645079751676107360, −7.63341759364011300648890703467, −7.24154424582590535084557023982, −5.86386211443646744963285926674, −5.26316474533697114425298586579, −4.64424522105575311157522917147, −3.52027701952687791433166308846, −2.61068377107659626706011855597, −0.19794344001578377881264968084, 1.22649099254439621590231990928, 2.22780394731086744065333156101, 3.04115792118313243600773856010, 4.17708986921298282895513197810, 4.79200119772352266437121717895, 6.31453186778697762654851795277, 7.07314477236039189637726255421, 7.62058378269002436420007915603, 8.462137997519310584669984052713, 9.407979235568723783188913889857

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