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Properties

Label 2-440-440.109-c0-0-0
Degree $2$
Conductor $440$
Sign $-0.707 - 0.707i$
Analytic cond. $0.219588$
Root an. cond. $0.468602$
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 − 9-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 + (0.707 − 0.707i)18-s + 1.00·20-s + (−0.707 − 0.707i)22-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 − 9-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 + (0.707 − 0.707i)18-s + 1.00·20-s + (−0.707 − 0.707i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(0.219588\)
Root analytic conductor: \(0.468602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4373932575\)
\(L(\frac12)\) \(\approx\) \(0.4373932575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 - iT \)
11 \( 1 - iT \)
good3 \( 1 + T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
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

−11.51640037055882422337455054672, −10.47291597980447218404415348406, −9.728629105698599038849830183115, −9.177630012185954439319359941488, −7.891845863095544999527609161898, −6.91475416948119669735322761033, −6.43531020284095794841359864086, −5.39996120862128113966806719757, −3.69815534626992734429555200618, −2.33010659161763654203787900882, 0.71796733386011223529485229546, 2.93468632043742506253612993293, 3.54960242554614991044676489987, 5.34741436691918161810374430745, 6.21631059534135387796936472067, 7.82255319516468315445188760218, 8.379771277873050611045285175078, 9.327219276288317311428240466926, 9.970142364196254564618461086188, 10.92631961480807810474456623716

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