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Properties

Label 2-3960-440.109-c0-0-4
Degree $2$
Conductor $3960$
Sign $0.707 - 0.707i$
Analytic cond. $1.97629$
Root an. cond. $1.40580$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + i·5-s + 1.41·7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + i·11-s + 1.41i·13-s + (−1.00 − 1.00i)14-s − 1.00·16-s + 1.41·17-s − 1.00·20-s + (0.707 − 0.707i)22-s − 25-s + (1.00 − 1.00i)26-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + i·5-s + 1.41·7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + i·11-s + 1.41i·13-s + (−1.00 − 1.00i)14-s − 1.00·16-s + 1.41·17-s − 1.00·20-s + (0.707 − 0.707i)22-s − 25-s + (1.00 − 1.00i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3960\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.97629\)
Root analytic conductor: \(1.40580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3960} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3960,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092895494\)
\(L(\frac12)\) \(\approx\) \(1.092895494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 - iT \)
11 \( 1 - iT \)
good7 \( 1 - 1.41T + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.780587617986931920600691956584, −7.988816694162300191061301765304, −7.32160880544288757699742539018, −6.99093144867118979164578867771, −5.79746747482598588582334750793, −4.67703488420814108205298399017, −4.08883051632040673056338765191, −3.10205122068549285682381847156, −2.04642552366789615559020104049, −1.56087167796996927417834091876, 0.878954898968080740854605256906, 1.52864835997380095223083001340, 2.97022727409248490972132171897, 4.25804450884861233973253832055, 5.16052783695257121279945934276, 5.50746978455129918063050764395, 6.17021967577763715367775231448, 7.58032594095621972620311549858, 7.85946271342171567073760242043, 8.409565045463735843561576414732

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