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Properties

Label 2-2240-40.29-c1-0-51
Degree $2$
Conductor $2240$
Sign $0.846 - 0.532i$
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  + 3.11·3-s + (0.660 + 2.13i)5-s + i·7-s + 6.70·9-s − 0.780i·11-s + 3.09·13-s + (2.05 + 6.65i)15-s − 7.31i·17-s − 0.207i·19-s + 3.11i·21-s + 1.52i·23-s + (−4.12 + 2.82i)25-s + 11.5·27-s + 3.60i·29-s + 7.94·31-s + ⋯
L(s)  = 1  + 1.79·3-s + (0.295 + 0.955i)5-s + 0.377i·7-s + 2.23·9-s − 0.235i·11-s + 0.857·13-s + (0.531 + 1.71i)15-s − 1.77i·17-s − 0.0476i·19-s + 0.679i·21-s + 0.318i·23-s + (−0.825 + 0.564i)25-s + 2.22·27-s + 0.669i·29-s + 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.887061939\)
\(L(\frac12)\) \(\approx\) \(3.887061939\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.660 - 2.13i)T \)
7 \( 1 - iT \)
good3 \( 1 - 3.11T + 3T^{2} \)
11 \( 1 + 0.780iT - 11T^{2} \)
13 \( 1 - 3.09T + 13T^{2} \)
17 \( 1 + 7.31iT - 17T^{2} \)
19 \( 1 + 0.207iT - 19T^{2} \)
23 \( 1 - 1.52iT - 23T^{2} \)
29 \( 1 - 3.60iT - 29T^{2} \)
31 \( 1 - 7.94T + 31T^{2} \)
37 \( 1 + 3.70T + 37T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 6.61T + 43T^{2} \)
47 \( 1 + 2.61iT - 47T^{2} \)
53 \( 1 + 8.69T + 53T^{2} \)
59 \( 1 - 14.6iT - 59T^{2} \)
61 \( 1 + 6.37iT - 61T^{2} \)
67 \( 1 - 7.21T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 0.502iT - 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−9.036468239994979284064396061063, −8.447675445205988942092677566248, −7.59994504783538767804611060655, −7.02141288166103109656568920161, −6.18916926642811938535547093109, −5.00039805637513882826133241379, −3.88203062330182885000474332296, −3.00368961928087653518417397466, −2.66979244037893389263184810228, −1.49071725599085498100560542481, 1.28726295993871255168934211146, 2.01148871658127960030773185239, 3.18134508266922037776049310162, 4.05136710740080593503231902381, 4.55424214477906679281470774823, 5.91980251084740537146978105914, 6.73400199963792186620814346160, 7.905195226701827634911491777083, 8.271758936354635953026626331539, 8.744787979552181302181167681761

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