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Properties

Label 2-2240-28.27-c1-0-59
Degree $2$
Conductor $2240$
Sign $-0.755 + 0.654i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  + 1.73·3-s + i·5-s + (−1.73 − 2i)7-s + 0.267i·11-s + 0.464i·13-s + 1.73i·15-s − 6.46i·17-s − 6·19-s + (−2.99 − 3.46i)21-s − 1.46i·23-s − 25-s − 5.19·27-s − 7.92·29-s − 6·31-s + 0.464i·33-s + ⋯
L(s)  = 1  + 1.00·3-s + 0.447i·5-s + (−0.654 − 0.755i)7-s + 0.0807i·11-s + 0.128i·13-s + 0.447i·15-s − 1.56i·17-s − 1.37·19-s + (−0.654 − 0.755i)21-s − 0.305i·23-s − 0.200·25-s − 1.00·27-s − 1.47·29-s − 1.07·31-s + 0.0807i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8477418654\)
\(L(\frac12)\) \(\approx\) \(0.8477418654\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 - 0.267iT - 11T^{2} \)
13 \( 1 - 0.464iT - 13T^{2} \)
17 \( 1 + 6.46iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 9.46T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 9.46iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 7.46iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 + 13.3iT - 97T^{2} \)
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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

−8.887744990885076882450898373049, −7.84672459810502548910562164830, −7.32830973633121985485286168168, −6.58217256337883346606065863023, −5.67525194776271474614683888872, −4.44039300845477006303835967867, −3.66677072509821674448823513781, −2.87098738860332728352054479488, −2.05101466215346182066550663208, −0.22766335112949957398417250885, 1.77202516625312643078175045202, 2.55993227511688063479947578651, 3.59314970336008862572585348162, 4.21994340056006105512705195320, 5.66588584184211456407633101647, 5.98520033957741246974910471758, 7.13121044925108297351342547441, 8.141851898283146868111630241156, 8.520298909053633238558095195655, 9.212248970566593225947262951186

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