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Properties

Label 2-440-55.32-c1-0-8
Degree $2$
Conductor $440$
Sign $0.867 - 0.498i$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
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  + (0.395 − 0.395i)3-s + (−0.0577 + 2.23i)5-s + (2.57 − 2.57i)7-s + 2.68i·9-s + (−2.35 + 2.33i)11-s + (1.81 + 1.81i)13-s + (0.861 + 0.906i)15-s + (2.63 − 2.63i)17-s + 4.84·19-s − 2.03i·21-s + (−0.648 + 0.648i)23-s + (−4.99 − 0.258i)25-s + (2.24 + 2.24i)27-s − 9.19·29-s + 9.88·31-s + ⋯
L(s)  = 1  + (0.228 − 0.228i)3-s + (−0.0258 + 0.999i)5-s + (0.973 − 0.973i)7-s + 0.895i·9-s + (−0.711 + 0.702i)11-s + (0.502 + 0.502i)13-s + (0.222 + 0.234i)15-s + (0.640 − 0.640i)17-s + 1.11·19-s − 0.444i·21-s + (−0.135 + 0.135i)23-s + (−0.998 − 0.0516i)25-s + (0.432 + 0.432i)27-s − 1.70·29-s + 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.867 - 0.498i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.867 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57412 + 0.420001i\)
\(L(\frac12)\) \(\approx\) \(1.57412 + 0.420001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.0577 - 2.23i)T \)
11 \( 1 + (2.35 - 2.33i)T \)
good3 \( 1 + (-0.395 + 0.395i)T - 3iT^{2} \)
7 \( 1 + (-2.57 + 2.57i)T - 7iT^{2} \)
13 \( 1 + (-1.81 - 1.81i)T + 13iT^{2} \)
17 \( 1 + (-2.63 + 2.63i)T - 17iT^{2} \)
19 \( 1 - 4.84T + 19T^{2} \)
23 \( 1 + (0.648 - 0.648i)T - 23iT^{2} \)
29 \( 1 + 9.19T + 29T^{2} \)
31 \( 1 - 9.88T + 31T^{2} \)
37 \( 1 + (-7.37 - 7.37i)T + 37iT^{2} \)
41 \( 1 + 5.37iT - 41T^{2} \)
43 \( 1 + (2.39 + 2.39i)T + 43iT^{2} \)
47 \( 1 + (2.29 + 2.29i)T + 47iT^{2} \)
53 \( 1 + (3.34 - 3.34i)T - 53iT^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + 6.46iT - 61T^{2} \)
67 \( 1 + (0.909 + 0.909i)T + 67iT^{2} \)
71 \( 1 + 5.95T + 71T^{2} \)
73 \( 1 + (1.99 + 1.99i)T + 73iT^{2} \)
79 \( 1 + 7.61T + 79T^{2} \)
83 \( 1 + (4.50 + 4.50i)T + 83iT^{2} \)
89 \( 1 + 2.85iT - 89T^{2} \)
97 \( 1 + (8.73 + 8.73i)T + 97iT^{2} \)
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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

−11.18819378352829878139984475994, −10.36500243241305180841162738742, −9.653067531674222647831544302787, −7.988411500017034340360795692229, −7.66136305977935553620871325271, −6.84408103622395720166572476001, −5.37927230145379078431147508640, −4.38017131694542961067917824099, −3.00943233916172564725123748592, −1.70283889212062445135910761577, 1.20120205205903350217117931867, 2.93039322086791116043567337079, 4.21896355791915262780665387735, 5.46504239150226161536045215466, 5.90826760819537261235964196255, 7.81434456813832294521737525333, 8.331921463217972829386760945875, 9.129354858330901908687464257054, 9.969821755056149225203410711870, 11.26673415726959587203885017822

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