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Properties

Label 2-1805-95.84-c0-0-1
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  + (0.707 − 1.22i)2-s + (−0.707 + 1.22i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.999 + 1.73i)6-s + (−0.499 − 0.866i)9-s + (−0.707 − 1.22i)10-s + 1.41·12-s + (0.707 + 1.22i)13-s + (0.707 + 1.22i)15-s + (0.499 − 0.866i)16-s − 1.41·18-s − 0.999·20-s + (−0.499 − 0.866i)25-s + 2·26-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.707 + 1.22i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.999 + 1.73i)6-s + (−0.499 − 0.866i)9-s + (−0.707 − 1.22i)10-s + 1.41·12-s + (0.707 + 1.22i)13-s + (0.707 + 1.22i)15-s + (0.499 − 0.866i)16-s − 1.41·18-s − 0.999·20-s + (−0.499 − 0.866i)25-s + 2·26-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: $\chi_{1805} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.527964791\)
\(L(\frac12)\) \(\approx\) \(1.527964791\)
\(L(1)\) 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 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.41T + T^{2} \)
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 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.584558832150710262453694530865, −9.105076786593589524895124520629, −8.026652247175540841132096890057, −6.60050261349348357503519818599, −5.64480108074740047496686458861, −5.01897227222610891168895507691, −4.22469595122607518637532847040, −3.85777897993267101066853400577, −2.45602797975260126361993665697, −1.31679355163291657121485635824, 1.34575692393145751466489546728, 2.70744949697791368295009133013, 3.90595830391495515150895757561, 5.21121631345087000205805225512, 5.88723929792624444368846182843, 6.27074304660760730083286023088, 6.98165101047589029290026821811, 7.65760307513148747016785159461, 8.182045537910466434136139412958, 9.491018825364395831617891192321

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