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Properties

Label 2-1805-5.2-c0-0-0
Degree $2$
Conductor $1805$
Sign $0.850 - 0.525i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·4-s − 5-s + (1 + i)7-s + i·9-s − 16-s + (1 + i)17-s + i·20-s + (−1 + i)23-s + 25-s + (1 − i)28-s + (−1 − i)35-s + 36-s + (1 − i)43-s − i·45-s + (1 + i)47-s + ⋯
L(s)  = 1  − i·4-s − 5-s + (1 + i)7-s + i·9-s − 16-s + (1 + i)17-s + i·20-s + (−1 + i)23-s + 25-s + (1 − i)28-s + (−1 − i)35-s + 36-s + (1 − i)43-s − i·45-s + (1 + i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.022509871\)
\(L(\frac12)\) \(\approx\) \(1.022509871\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.489092204155622990430423793002, −8.642239569006954295654568838404, −7.948454893028368274881781531305, −7.43360453773656571025037337857, −6.07271446253608712884737214147, −5.46165177080377468391558786678, −4.77864539512211053608343347480, −3.82621799960643793626928947285, −2.38076693760357818348144392160, −1.47383051983270950004399722448, 0.860885543085242964320735710058, 2.67587418014538155061318447713, 3.74933087467649909020755424101, 4.15038990101714729844658516666, 5.06047125634283692279483746367, 6.48767767785950384848932944566, 7.29628909720040840839586411009, 7.77426947829995868734140954650, 8.384568504675122198166225128128, 9.221163543707356833542591620325

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