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Properties

Label 2-1665-185.73-c0-0-3
Degree $2$
Conductor $1665$
Sign $-0.731 - 0.681i$
Analytic cond. $0.830943$
Root an. cond. $0.911560$
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  + (1.17 + 1.17i)2-s + 1.76i·4-s + (0.555 + 0.831i)5-s + (−0.541 + 0.541i)7-s + (−0.899 + 0.899i)8-s + (−0.324 + 1.63i)10-s − 1.27·14-s − 0.351·16-s + (−0.785 − 0.785i)17-s + (−1.46 + 0.980i)20-s + (1.38 − 1.38i)23-s + (−0.382 + 0.923i)25-s + (−0.955 − 0.955i)28-s − 1.96·29-s + (0.487 + 0.487i)32-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)2-s + 1.76i·4-s + (0.555 + 0.831i)5-s + (−0.541 + 0.541i)7-s + (−0.899 + 0.899i)8-s + (−0.324 + 1.63i)10-s − 1.27·14-s − 0.351·16-s + (−0.785 − 0.785i)17-s + (−1.46 + 0.980i)20-s + (1.38 − 1.38i)23-s + (−0.382 + 0.923i)25-s + (−0.955 − 0.955i)28-s − 1.96·29-s + (0.487 + 0.487i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1665\)    =    \(3^{2} \cdot 5 \cdot 37\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(0.830943\)
Root analytic conductor: \(0.911560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1665} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1665,\ (\ :0),\ -0.731 - 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.104125585\)
\(L(\frac12)\) \(\approx\) \(2.104125585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.555 - 0.831i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
7 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
29 \( 1 + 1.96T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.390T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.66T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.592441468787650219609709230445, −9.047680406646094408133788789787, −7.928231507021396082973895976804, −6.99468026415818237891831331118, −6.67002377336148348395692641899, −5.82101981224283897495226427575, −5.19795116624965153298212099325, −4.19198185826713134873313941896, −3.16277638751600892054543553552, −2.39111232059209438387534369115, 1.25521215594185721436674110902, 2.19963494788065487685176693273, 3.40357311677783989689668869056, 4.07061139837961740183818798076, 4.99545518308153569611148309110, 5.63895367256931147226324951693, 6.50513715332528854765805330391, 7.60096547530203164723382815645, 8.781416475349803292923366414078, 9.564604372475551877933204712440

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