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Properties

Label 2-2664-37.10-c1-0-5
Degree $2$
Conductor $2664$
Sign $-0.648 - 0.761i$
Analytic cond. $21.2721$
Root an. cond. $4.61217$
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  + (−1.87 − 3.24i)5-s + (2.03 + 3.52i)7-s − 1.66·11-s + (0.106 + 0.184i)13-s + (−0.5 + 0.866i)17-s + (−0.271 − 0.470i)19-s + 4.61·23-s + (−4.51 + 7.82i)25-s − 9.65·29-s − 1.66·31-s + (7.62 − 13.2i)35-s + (−3.22 − 5.16i)37-s + (−0.729 − 1.26i)41-s − 2.74·43-s − 3.89·47-s + ⋯
L(s)  = 1  + (−0.837 − 1.45i)5-s + (0.769 + 1.33i)7-s − 0.503·11-s + (0.0295 + 0.0511i)13-s + (−0.121 + 0.210i)17-s + (−0.0623 − 0.107i)19-s + 0.962·23-s + (−0.903 + 1.56i)25-s − 1.79·29-s − 0.299·31-s + (1.28 − 2.23i)35-s + (−0.529 − 0.848i)37-s + (−0.113 − 0.197i)41-s − 0.418·43-s − 0.568·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.648 - 0.761i$
Analytic conductor: \(21.2721\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :1/2),\ -0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4119416710\)
\(L(\frac12)\) \(\approx\) \(0.4119416710\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 + (3.22 + 5.16i)T \)
good5 \( 1 + (1.87 + 3.24i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.03 - 3.52i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
13 \( 1 + (-0.106 - 0.184i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.271 + 0.470i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 + 9.65T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
41 \( 1 + (0.729 + 1.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + 3.89T + 47T^{2} \)
53 \( 1 + (3.86 - 6.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.563 + 0.975i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.85 - 4.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.01 + 5.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.49 - 6.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 + (-4.77 - 8.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.22 - 10.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.72 - 4.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.99T + 97T^{2} \)
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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

−9.068705279174907420784985276124, −8.345281669393610167276172889117, −7.922278261798383152107200968022, −7.00591261318680721115560553392, −5.59581235284750078582272780976, −5.32625541770788964693322956346, −4.54169240213270974247844912926, −3.63475475489356609995755885780, −2.35089800590800378149833593014, −1.36123029326194816673563989172, 0.13537316502969334761080629619, 1.71848423897861483772371747724, 3.02385409837256913777035877277, 3.65257745328208434445906413192, 4.47941103392809050505712006633, 5.38119438952972808538621025543, 6.65820431231124756928545820751, 7.08934960012281132223244585264, 7.73618134472575958645295804138, 8.223499772622360641345494103932

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