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Properties

Label 2-2240-140.139-c1-0-45
Degree $2$
Conductor $2240$
Sign $0.956 + 0.290i$
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  + 3.06i·3-s + (−1.39 + 1.75i)5-s + (−2.46 + 0.972i)7-s − 6.38·9-s − 5.46i·11-s − 1.22·13-s + (−5.36 − 4.25i)15-s − 0.813·17-s + 6.68·19-s + (−2.97 − 7.53i)21-s − 3.51·23-s + (−1.13 − 4.86i)25-s − 10.3i·27-s + 8.09·29-s − 9.00·31-s + ⋯
L(s)  = 1  + 1.76i·3-s + (−0.621 + 0.783i)5-s + (−0.930 + 0.367i)7-s − 2.12·9-s − 1.64i·11-s − 0.340·13-s + (−1.38 − 1.09i)15-s − 0.197·17-s + 1.53·19-s + (−0.650 − 1.64i)21-s − 0.733·23-s + (−0.227 − 0.973i)25-s − 1.99i·27-s + 1.50·29-s − 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4355747521\)
\(L(\frac12)\) \(\approx\) \(0.4355747521\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 1.75i)T \)
7 \( 1 + (2.46 - 0.972i)T \)
good3 \( 1 - 3.06iT - 3T^{2} \)
11 \( 1 + 5.46iT - 11T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 + 0.813T + 17T^{2} \)
19 \( 1 - 6.68T + 19T^{2} \)
23 \( 1 + 3.51T + 23T^{2} \)
29 \( 1 - 8.09T + 29T^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 - 4.61iT - 37T^{2} \)
41 \( 1 - 2.59iT - 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + 5.97iT - 47T^{2} \)
53 \( 1 + 1.51iT - 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 3.16iT - 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 + 2.19iT - 79T^{2} \)
83 \( 1 - 7.88iT - 83T^{2} \)
89 \( 1 + 1.19iT - 89T^{2} \)
97 \( 1 + 0.813T + 97T^{2} \)
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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

−9.110342934059313833621004031732, −8.489543739165321229924324603174, −7.57507089116436187009062756669, −6.41014213275618298095276536850, −5.78854308282954567199689935401, −4.95805886646561174204541052030, −3.89196358124311406395833761043, −3.24887567860662107222826825871, −2.87177893458879355548448782392, −0.17696085861910068119095999876, 1.01232194159531938608973643600, 1.99734111263524930551632047507, 3.07363007302446205808206244890, 4.21899890682688960786013658686, 5.22581964954774189317941221436, 6.13866084113249105298497399912, 7.07821735184439414701169367367, 7.40662161485344019540695657379, 7.938605249891043575532164430005, 9.051101091386206201672607520745

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