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Properties

Label 2-2240-40.29-c1-0-65
Degree $2$
Conductor $2240$
Sign $0.178 + 0.983i$
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  + 2.05·3-s + (1.27 − 1.83i)5-s + i·7-s + 1.24·9-s − 5.33i·11-s − 0.785·13-s + (2.62 − 3.78i)15-s − 1.61i·17-s − 4.54i·19-s + 2.05i·21-s + 7.09i·23-s + (−1.75 − 4.68i)25-s − 3.62·27-s − 7.74i·29-s + 4.11·31-s + ⋯
L(s)  = 1  + 1.18·3-s + (0.569 − 0.822i)5-s + 0.377i·7-s + 0.413·9-s − 1.60i·11-s − 0.217·13-s + (0.676 − 0.977i)15-s − 0.392i·17-s − 1.04i·19-s + 0.449i·21-s + 1.47i·23-s + (−0.351 − 0.936i)25-s − 0.697·27-s − 1.43i·29-s + 0.739·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.178 + 0.983i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.178 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.722752747\)
\(L(\frac12)\) \(\approx\) \(2.722752747\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 + 1.83i)T \)
7 \( 1 - iT \)
good3 \( 1 - 2.05T + 3T^{2} \)
11 \( 1 + 5.33iT - 11T^{2} \)
13 \( 1 + 0.785T + 13T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 + 4.54iT - 19T^{2} \)
23 \( 1 - 7.09iT - 23T^{2} \)
29 \( 1 + 7.74iT - 29T^{2} \)
31 \( 1 - 4.11T + 31T^{2} \)
37 \( 1 - 0.480T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 2.00T + 43T^{2} \)
47 \( 1 - 2.24iT - 47T^{2} \)
53 \( 1 + 5.57T + 53T^{2} \)
59 \( 1 + 5.02iT - 59T^{2} \)
61 \( 1 - 4.55iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 2.00iT - 73T^{2} \)
79 \( 1 + 6.61T + 79T^{2} \)
83 \( 1 - 8.91T + 83T^{2} \)
89 \( 1 - 0.428T + 89T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−8.885539467802142126526811874499, −8.234190456462360539781168060628, −7.69483204881138226861296711324, −6.41535298364201696337132129971, −5.68241106199471552862404316247, −4.91472101897854597049566748842, −3.76636805391083066889732736326, −2.91051509131022506911614357430, −2.14542622047763174825906926865, −0.77484950596102852030747036810, 1.72493627193917410591681847926, 2.40391553191293171297846619487, 3.30407389426621610701583984424, 4.17142175625035490534321916520, 5.14229064424103247420629464912, 6.34533623782416903147911780901, 6.96500021615220625635286731228, 7.70657685484655111401675797757, 8.383998510558349580260354084480, 9.273902621623776387119727856152

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