Properties

Label 2-440-5.4-c1-0-6
Degree $2$
Conductor $440$
Sign $0.981 + 0.189i$
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  − 0.339i·3-s + (0.422 − 2.19i)5-s + 4.05i·7-s + 2.88·9-s + 11-s + 4i·13-s + (−0.744 − 0.143i)15-s − 7.74i·17-s + 7.06·19-s + 1.37·21-s − 2.72i·23-s + (−4.64 − 1.85i)25-s − 1.99i·27-s + 4.73·29-s + 0.219·31-s + ⋯
L(s)  = 1  − 0.195i·3-s + (0.189 − 0.981i)5-s + 1.53i·7-s + 0.961·9-s + 0.301·11-s + 1.10i·13-s + (−0.192 − 0.0370i)15-s − 1.87i·17-s + 1.62·19-s + 0.299·21-s − 0.568i·23-s + (−0.928 − 0.371i)25-s − 0.384i·27-s + 0.878·29-s + 0.0394·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57069 - 0.149840i\)
\(L(\frac12)\) \(\approx\) \(1.57069 - 0.149840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.422 + 2.19i)T \)
11 \( 1 - T \)
good3 \( 1 + 0.339iT - 3T^{2} \)
7 \( 1 - 4.05iT - 7T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 7.74iT - 17T^{2} \)
19 \( 1 - 7.06T + 19T^{2} \)
23 \( 1 + 2.72iT - 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 0.219T + 31T^{2} \)
37 \( 1 + 1.32iT - 37T^{2} \)
41 \( 1 + 7.79T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 - 3.01iT - 47T^{2} \)
53 \( 1 - 5.03iT - 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 4.70iT - 67T^{2} \)
71 \( 1 + 2.52T + 71T^{2} \)
73 \( 1 + 4.10iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 3.63iT - 83T^{2} \)
89 \( 1 + 9.88T + 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−11.55252019981556757871934285006, −9.715058074134019941643398740076, −9.404739589239510252969584607955, −8.564565387095456171290450199669, −7.40985857280803417662108222796, −6.38137158682043594600056988830, −5.21802359534506346870426895285, −4.54389908106623049397871124333, −2.75740899915465606808004691186, −1.39436105128766359100462172793, 1.38300258942287218742815154111, 3.36211292532851954184092710292, 4.00251110926346233435479728813, 5.45985744255461764364665984868, 6.72097054005276740558471037699, 7.32762277919823656530048678379, 8.204298656903447153848531261720, 9.847181917699469309229198767843, 10.26183303305859703767457684285, 10.79368637468191032377830009653

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