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Properties

Label 2-1805-95.69-c0-0-2
Degree $2$
Conductor $1805$
Sign $-0.671 + 0.740i$
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  + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)9-s − 2·11-s + (−0.499 − 0.866i)16-s − 0.999·20-s + (−0.499 + 0.866i)25-s + (−0.499 − 0.866i)36-s + (−1 + 1.73i)44-s − 0.999·45-s + 49-s + (1 + 1.73i)55-s + (1 − 1.73i)61-s − 0.999·64-s + (−0.499 + 0.866i)80-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)9-s − 2·11-s + (−0.499 − 0.866i)16-s − 0.999·20-s + (−0.499 + 0.866i)25-s + (−0.499 − 0.866i)36-s + (−1 + 1.73i)44-s − 0.999·45-s + 49-s + (1 + 1.73i)55-s + (1 − 1.73i)61-s − 0.999·64-s + (−0.499 + 0.866i)80-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9051750357\)
\(L(\frac12)\) \(\approx\) \(0.9051750357\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.284233658200396951443201340477, −8.343703000118279132387933579893, −7.59717800370047158160619429912, −6.84912811141498521671640861474, −5.78470530837588323895847250057, −5.19030032299635478293281023225, −4.37573129061761763166196690533, −3.15057363362349909747451420927, −1.97808180144457881941174612259, −0.63738001368318639160124004468, 2.25804372038258706175905864861, 2.77320480571584977700272445276, 3.80210848357441750356037224391, 4.77737323369318287143758728101, 5.76501086707982901293560452799, 6.89272211029900109895360237539, 7.52261717655196593223978452712, 7.902099249505173703172214692599, 8.662222920761846503929618367783, 10.10194378396398369010649243791

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