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Properties

Label 2-1805-95.79-c0-0-2
Degree $2$
Conductor $1805$
Sign $-0.486 + 0.873i$
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  + (−0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (−0.766 + 0.642i)9-s + (1 − 1.73i)11-s + (−0.939 + 0.342i)16-s − 0.999·20-s + (−0.939 − 0.342i)25-s + (0.766 + 0.642i)36-s + (−1.87 − 0.684i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (−1.53 − 1.28i)55-s + (−0.347 − 1.96i)61-s + (0.5 + 0.866i)64-s + (0.173 + 0.984i)80-s + (0.173 − 0.984i)81-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (−0.766 + 0.642i)9-s + (1 − 1.73i)11-s + (−0.939 + 0.342i)16-s − 0.999·20-s + (−0.939 − 0.342i)25-s + (0.766 + 0.642i)36-s + (−1.87 − 0.684i)44-s + (0.5 + 0.866i)45-s + (−0.5 + 0.866i)49-s + (−1.53 − 1.28i)55-s + (−0.347 − 1.96i)61-s + (0.5 + 0.866i)64-s + (0.173 + 0.984i)80-s + (0.173 − 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.486 + 0.873i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9984921943\)
\(L(\frac12)\) \(\approx\) \(0.9984921943\)
\(L(1)\) 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 \( 1 + (0.173 + 0.984i)T^{2} \)
3 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.347 + 1.96i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T^{2} \)
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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

−9.082309274772167831092538854290, −8.633180430065631128441500460816, −7.896522945823823665570372767147, −6.46073406993987387905853927016, −5.90384495324182815327834882066, −5.26335767467871389704472840594, −4.42752197836894036904485195778, −3.28114446003872379141281183488, −1.87982706269574459843208127776, −0.76050564590831032545194604159, 1.99502394038673243258657916992, 2.98002494942958513666186367721, 3.80107478388469213971678224770, 4.58894661660637012786893463397, 5.88012017478560643630940827857, 6.83266836266993455369051582788, 7.15487689952754025641777283651, 8.107388456198378609656516383410, 9.036126304009037904032186067303, 9.585020750931402238229404186878

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