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Properties

Label 2-3330-5.4-c1-0-17
Degree $2$
Conductor $3330$
Sign $-0.493 - 0.869i$
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  − i·2-s − 4-s + (1.10 + 1.94i)5-s + 3.51i·7-s + i·8-s + (1.94 − 1.10i)10-s + 0.290·11-s + 7.12i·13-s + 3.51·14-s + 16-s − 6.17i·17-s − 5.83·19-s + (−1.10 − 1.94i)20-s − 0.290i·22-s + 6.45i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.493 + 0.869i)5-s + 1.32i·7-s + 0.353i·8-s + (0.614 − 0.349i)10-s + 0.0874·11-s + 1.97i·13-s + 0.938·14-s + 0.250·16-s − 1.49i·17-s − 1.33·19-s + (−0.246 − 0.434i)20-s − 0.0618i·22-s + 1.34i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.493 - 0.869i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.263535934\)
\(L(\frac12)\) \(\approx\) \(1.263535934\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-1.10 - 1.94i)T \)
37 \( 1 - iT \)
good7 \( 1 - 3.51iT - 7T^{2} \)
11 \( 1 - 0.290T + 11T^{2} \)
13 \( 1 - 7.12iT - 13T^{2} \)
17 \( 1 + 6.17iT - 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 - 6.45iT - 23T^{2} \)
29 \( 1 + 1.18T + 29T^{2} \)
31 \( 1 - 9.77T + 31T^{2} \)
41 \( 1 + 1.64T + 41T^{2} \)
43 \( 1 + 5.34iT - 43T^{2} \)
47 \( 1 - 5.69iT - 47T^{2} \)
53 \( 1 + 9.32iT - 53T^{2} \)
59 \( 1 - 5.94T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 + 6.04iT - 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 1.71iT - 73T^{2} \)
79 \( 1 + 8.25T + 79T^{2} \)
83 \( 1 + 2.41iT - 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−9.098040412200783212221177222854, −8.428718049505140293764140512275, −7.24695518992823222523010679948, −6.58655261124235981900265037841, −5.88463505769287615659080165178, −4.99150701302129643027303269590, −4.16063542725862511421712535374, −3.05460832599774294740717215629, −2.34360815015939876323082081045, −1.71066802611428041004519689702, 0.38784780962236227294971341510, 1.30977115654791018736151112033, 2.78222814885923156315006550577, 4.10092364083832935779610141481, 4.42125956970159787148900008647, 5.47463083762556223578011550712, 6.14257646441993188645624060767, 6.76904637974825703217787310107, 7.86941092268655470084706807107, 8.260301817752924316308767439198

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