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Properties

Label 2-3330-185.184-c1-0-24
Degree $2$
Conductor $3330$
Sign $-0.0485 - 0.998i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (−1.28 + 1.83i)5-s + 2.06i·7-s − 8-s + (1.28 − 1.83i)10-s − 3.77·11-s + 2.88·13-s − 2.06i·14-s + 16-s + 5.80·17-s − 0.157i·19-s + (−1.28 + 1.83i)20-s + 3.77·22-s + 5.41·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.573 + 0.819i)5-s + 0.779i·7-s − 0.353·8-s + (0.405 − 0.579i)10-s − 1.13·11-s + 0.799·13-s − 0.551i·14-s + 0.250·16-s + 1.40·17-s − 0.0360i·19-s + (−0.286 + 0.409i)20-s + 0.805·22-s + 1.12·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.0485 - 0.998i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.0485 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.101518307\)
\(L(\frac12)\) \(\approx\) \(1.101518307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (1.28 - 1.83i)T \)
37 \( 1 + (-4.80 + 3.72i)T \)
good7 \( 1 - 2.06iT - 7T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 + 0.157iT - 19T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 - 4.29iT - 29T^{2} \)
31 \( 1 - 0.425iT - 31T^{2} \)
41 \( 1 + 0.923T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 0.676iT - 47T^{2} \)
53 \( 1 + 9.87iT - 53T^{2} \)
59 \( 1 - 8.47iT - 59T^{2} \)
61 \( 1 - 1.23iT - 61T^{2} \)
67 \( 1 - 6.45iT - 67T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 + 0.980iT - 73T^{2} \)
79 \( 1 - 8.04iT - 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.696129621309039120418698716982, −8.090601315588780023362248841647, −7.45795505469539746849071782336, −6.80192475716227056030484834456, −5.79528871890688764743580002004, −5.30805249488982042824835021099, −3.93188246242398765278165086185, −3.01268758601997788313515111768, −2.45844102472987997676545061559, −0.992959746731061415074735908999, 0.57081030389737276870736455963, 1.31913512405048008169892507749, 2.78455701631486447916364491464, 3.66706132450371232962561523021, 4.57324608993701239745000492143, 5.43483592033693100841288248213, 6.20154043897514188435584516155, 7.40967268889090755404749419987, 7.70250365081322479955818124846, 8.326822566989869839970994543747

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