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Properties

Label 2-2240-280.277-c0-0-2
Degree $2$
Conductor $2240$
Sign $0.956 + 0.290i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)3-s + (−0.965 − 0.258i)5-s + (0.707 + 0.707i)7-s + (0.866 − 0.5i)9-s + (0.866 + 0.5i)11-s + (−0.707 − 0.707i)13-s − 1.41·15-s + (−0.5 − 0.866i)19-s + (1.22 + 0.707i)21-s + (0.965 + 0.258i)23-s + (0.866 + 0.499i)25-s + 1.41·29-s + (0.707 − 1.22i)31-s + (1.36 + 0.366i)33-s + (−0.500 − 0.866i)35-s + ⋯
L(s)  = 1  + (1.36 − 0.366i)3-s + (−0.965 − 0.258i)5-s + (0.707 + 0.707i)7-s + (0.866 − 0.5i)9-s + (0.866 + 0.5i)11-s + (−0.707 − 0.707i)13-s − 1.41·15-s + (−0.5 − 0.866i)19-s + (1.22 + 0.707i)21-s + (0.965 + 0.258i)23-s + (0.866 + 0.499i)25-s + 1.41·29-s + (0.707 − 1.22i)31-s + (1.36 + 0.366i)33-s + (−0.500 − 0.866i)35-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.757123181\)
\(L(\frac12)\) \(\approx\) \(1.757123181\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.021011894865317524639829284714, −8.359549087364715266246244973858, −7.80498468971331996058573763493, −7.22650384465936819049157639643, −6.23338213289858797788248847714, −4.82170079110954686252092153632, −4.42940273900140448301864617860, −3.13060545303501495221070449148, −2.58920196728580531845538035283, −1.35544173830992074697060118352, 1.42184147963846898914523592432, 2.75630991348065946169831666608, 3.51710310865447464983579726563, 4.28055493696478398158568074210, 4.80402900896928506494614870743, 6.47066987866771911541175107251, 7.10958264520534036100820567510, 7.929286316013831075788731362543, 8.498454607601325677622236863548, 8.960436201754652682336314167385

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