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Properties

Label 2-3330-37.36-c1-0-38
Degree $2$
Conductor $3330$
Sign $0.986 - 0.164i$
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 + i·5-s + 7-s − i·8-s − 10-s − 5·11-s + i·13-s + i·14-s + 16-s − 3i·17-s + i·19-s − i·20-s − 5i·22-s − 5i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.377·7-s − 0.353i·8-s − 0.316·10-s − 1.50·11-s + 0.277i·13-s + 0.267i·14-s + 0.250·16-s − 0.727i·17-s + 0.229i·19-s − 0.223i·20-s − 1.06i·22-s − 1.04i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (2071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.356337635\)
\(L(\frac12)\) \(\approx\) \(1.356337635\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
37 \( 1 + (-6 + i)T \)
good7 \( 1 - T + 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 - iT - 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−8.364672554672545536394830854336, −7.84210593465563758868825123247, −7.28894248506984663678851539113, −6.37394297264298744038576319345, −5.72512554712302053034666737638, −4.85958482448872533986596414630, −4.25090127148647929878864459534, −2.98078241499529770830956438319, −2.23771745466128998460607799347, −0.51777560622550167115141344598, 0.916219283107556203928735078368, 2.04391552724843100561174529793, 2.90266843032164941989041034959, 3.86998340534345196321746900518, 4.77537439400466608574764601604, 5.39880402855016171522091088185, 6.09858532968042604629183869045, 7.60557394179398900670847159036, 7.73002553865728711709980732007, 8.777185790020573194376333798208

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