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Properties

Label 2-1805-95.92-c0-0-0
Degree $2$
Conductor $1805$
Sign $0.979 - 0.203i$
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  + (0.984 − 0.173i)4-s + (−0.173 + 0.984i)5-s + (0.366 − 1.36i)7-s + (0.642 + 0.766i)9-s + (0.939 − 0.342i)16-s + (0.123 + 1.40i)17-s + 0.999i·20-s + (−1.15 − 0.811i)23-s + (−0.939 − 0.342i)25-s + (0.123 − 1.40i)28-s + (1.28 + 0.597i)35-s + (0.766 + 0.642i)36-s + (−0.811 − 1.15i)43-s + (−0.866 + 0.5i)45-s + (1.40 + 0.123i)47-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)4-s + (−0.173 + 0.984i)5-s + (0.366 − 1.36i)7-s + (0.642 + 0.766i)9-s + (0.939 − 0.342i)16-s + (0.123 + 1.40i)17-s + 0.999i·20-s + (−1.15 − 0.811i)23-s + (−0.939 − 0.342i)25-s + (0.123 − 1.40i)28-s + (1.28 + 0.597i)35-s + (0.766 + 0.642i)36-s + (−0.811 − 1.15i)43-s + (−0.866 + 0.5i)45-s + (1.40 + 0.123i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504054552\)
\(L(\frac12)\) \(\approx\) \(1.504054552\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T^{2} \)
3 \( 1 + (-0.642 - 0.766i)T^{2} \)
7 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.642 - 0.766i)T^{2} \)
17 \( 1 + (-0.123 - 1.40i)T + (-0.984 + 0.173i)T^{2} \)
23 \( 1 + (1.15 + 0.811i)T + (0.342 + 0.939i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.811 + 1.15i)T + (-0.342 + 0.939i)T^{2} \)
47 \( 1 + (-1.40 - 0.123i)T + (0.984 + 0.173i)T^{2} \)
53 \( 1 + (0.342 + 0.939i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.984 + 0.173i)T^{2} \)
71 \( 1 + (-0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.597 - 1.28i)T + (-0.642 - 0.766i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.984 + 0.173i)T^{2} \)
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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

−10.06993987311620162375081790228, −8.357968579792882746996290292651, −7.70200016513990149551358768597, −7.15382971331045031773903109850, −6.51494074866805702071614236815, −5.65028625064829296495606467538, −4.28784621775159118071403915721, −3.71052040635492210571213716742, −2.42468792117617895470378742576, −1.53763349035060186651787064285, 1.39083430413202575321031520525, 2.38414033580703548605117444181, 3.45733120901362629869628891351, 4.58017138474897868507621587648, 5.51806018875588513767857566137, 6.11194032812078470057404053219, 7.16658005977438615643548189096, 7.85606328850621023167736643437, 8.666934631074748140107360013635, 9.397665847503512469862224965916

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