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Properties

Label 2-330-55.17-c1-0-10
Degree $2$
Conductor $330$
Sign $0.695 + 0.718i$
Analytic cond. $2.63506$
Root an. cond. $1.62328$
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  + (0.891 + 0.453i)2-s + (−0.987 + 0.156i)3-s + (0.587 + 0.809i)4-s + (−0.169 − 2.22i)5-s + (−0.951 − 0.309i)6-s + (0.617 − 3.89i)7-s + (0.156 + 0.987i)8-s + (0.951 − 0.309i)9-s + (0.861 − 2.06i)10-s + (−2.89 − 1.62i)11-s + (−0.707 − 0.707i)12-s + (1.51 − 2.98i)13-s + (2.31 − 3.19i)14-s + (0.515 + 2.17i)15-s + (−0.309 + 0.951i)16-s + (2.59 + 5.08i)17-s + ⋯
L(s)  = 1  + (0.630 + 0.321i)2-s + (−0.570 + 0.0903i)3-s + (0.293 + 0.404i)4-s + (−0.0756 − 0.997i)5-s + (−0.388 − 0.126i)6-s + (0.233 − 1.47i)7-s + (0.0553 + 0.349i)8-s + (0.317 − 0.103i)9-s + (0.272 − 0.652i)10-s + (−0.872 − 0.488i)11-s + (−0.204 − 0.204i)12-s + (0.421 − 0.827i)13-s + (0.619 − 0.853i)14-s + (0.133 + 0.561i)15-s + (−0.0772 + 0.237i)16-s + (0.628 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(330\)    =    \(2 \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(2.63506\)
Root analytic conductor: \(1.62328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{330} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 330,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39147 - 0.589866i\)
\(L(\frac12)\) \(\approx\) \(1.39147 - 0.589866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 + (0.169 + 2.22i)T \)
11 \( 1 + (2.89 + 1.62i)T \)
good7 \( 1 + (-0.617 + 3.89i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-1.51 + 2.98i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-2.59 - 5.08i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.07 - 1.50i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.92 + 2.92i)T - 23iT^{2} \)
29 \( 1 + (-0.635 + 0.461i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.973 + 2.99i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-10.0 - 1.59i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (6.69 - 9.21i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (4.78 + 4.78i)T + 43iT^{2} \)
47 \( 1 + (-1.59 - 10.0i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (3.91 + 1.99i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (6.68 + 9.19i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.71 - 0.881i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.47 - 7.47i)T + 67iT^{2} \)
71 \( 1 + (1.24 - 3.82i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-11.8 - 1.87i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.18 + 3.63i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.60 + 1.83i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
97 \( 1 + (-2.95 + 5.79i)T + (-57.0 - 78.4i)T^{2} \)
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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

−11.44488937494485287147083385883, −10.69156514584574602402453703413, −9.858496445155467997628492449656, −8.140807226896515029517849415257, −7.83440037612738447649924541690, −6.37331374145092164382810937033, −5.41449943824117029445929552689, −4.52139764917251276813637982340, −3.49385227253392673969623338124, −1.02062669826793631634533344674, 2.15837028854017282982411671076, 3.20422733121004750087221715071, 4.91410772307303022115974120803, 5.62289961392930920917024172514, 6.71153321991478743423142312498, 7.61440705537282884063334720084, 9.129347573925596802265136712731, 10.04696267517948032066193769093, 11.14930638446828295700209486842, 11.66967664219707295847511101299

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