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Properties

Label 2-1805-95.59-c0-0-1
Degree $2$
Conductor $1805$
Sign $0.934 + 0.356i$
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  + (0.939 + 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.173 − 0.984i)9-s + (1 − 1.73i)11-s + (0.766 + 0.642i)16-s − 20-s + (0.766 − 0.642i)25-s + (0.173 − 0.984i)36-s + (1.53 − 1.28i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (−0.347 + 1.96i)55-s + (1.87 + 0.684i)61-s + (0.500 + 0.866i)64-s + (−0.939 − 0.342i)80-s + (−0.939 + 0.342i)81-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)4-s + (−0.939 + 0.342i)5-s + (−0.173 − 0.984i)9-s + (1 − 1.73i)11-s + (0.766 + 0.642i)16-s − 20-s + (0.766 − 0.642i)25-s + (0.173 − 0.984i)36-s + (1.53 − 1.28i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (−0.347 + 1.96i)55-s + (1.87 + 0.684i)61-s + (0.500 + 0.866i)64-s + (−0.939 − 0.342i)80-s + (−0.939 + 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.934 + 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256326172\)
\(L(\frac12)\) \(\approx\) \(1.256326172\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T^{2} \)
3 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.87 - 0.684i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T^{2} \)
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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

−9.226866457780885609149043900824, −8.517439331773773822211621526116, −7.896049966513681869691897202696, −6.88128931310006562275676072200, −6.43804552585346934447718078479, −5.65428764236974543620038661643, −4.04883269614687562857412384411, −3.49275855474324674702688405565, −2.77956927167641606079338957195, −1.05990217004758937017361241634, 1.50995150883724228360282460664, 2.40970250396672848062314110143, 3.70558103905421531082912955313, 4.63244764492594870151805979057, 5.33750336412210329528675190446, 6.57056498811158570249991166592, 7.16732812334231693788432522025, 7.76959870443401389935655513046, 8.609225270588778728887401822860, 9.650385640120276749075800718007

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