Properties

Label 2-2240-5.4-c1-0-8
Degree $2$
Conductor $2240$
Sign $-0.447 - 0.894i$
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  i·3-s + (−2 + i)5-s + i·7-s + 2·9-s + 11-s + i·13-s + (1 + 2i)15-s + 3i·17-s − 4·19-s + 21-s + 2i·23-s + (3 − 4i)25-s − 5i·27-s − 29-s − 6·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.894 + 0.447i)5-s + 0.377i·7-s + 0.666·9-s + 0.301·11-s + 0.277i·13-s + (0.258 + 0.516i)15-s + 0.727i·17-s − 0.917·19-s + 0.218·21-s + 0.417i·23-s + (0.600 − 0.800i)25-s − 0.962i·27-s − 0.185·29-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7546741021\)
\(L(\frac12)\) \(\approx\) \(0.7546741021\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
7 \( 1 - iT \)
good3 \( 1 + iT - 3T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 19iT - 97T^{2} \)
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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

−9.093539465799854060103428884620, −8.417383794736578545521292719953, −7.68793383710606547363503198831, −6.93655542602082511837804887232, −6.45597334346965511796262917175, −5.39078557897195606851427934222, −4.22736966864715661688539858549, −3.69132277762757181282166305959, −2.43008637385855123559522439640, −1.41627846007304574185549093111, 0.26782730409805383087605093350, 1.67886796995363066051158480688, 3.22355143435925611147062859124, 3.98573084281879101764207324267, 4.62272669291462483041148488468, 5.35621408071497572020905633756, 6.64417896302252110208382440785, 7.24982985865202971765063356171, 8.058417322185325920710096687510, 8.817657413218112451353241511906

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