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Properties

Label 2-440-440.109-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(1.209767891\)
\(L(\frac12)\) \(\approx\) \(1.209767891\)
\(L(1)\) 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

−11.47969916565384242050940438057, −11.13522895612201350728807069233, −9.016037064244885667602036812569, −8.679763585796626946055929799999, −7.931177600401032653747596790483, −6.65084152062920225651943853148, −5.59624801057611809939346454741, −4.81899530931136962535141850816, −3.95614772663941259388653209253, −2.16076193369303911286801865294, 2.04966231137351557033717291949, 2.99994336543925368131594574345, 4.40305546739481348527670516412, 5.31334382028543013068895952829, 6.32986468732004350411516088296, 7.51006007044356890948305602124, 8.513838040359058416542435159128, 9.772221938777555635619963067982, 10.76366657859371949396651036320, 11.14527067575700679556307433799

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