Properties

Label 2-2240-8.5-c1-0-12
Degree $2$
Conductor $2240$
Sign $-0.707 - 0.707i$
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  + 2.29i·3-s i·5-s − 7-s − 2.25·9-s + 0.744i·11-s + 3.83i·13-s + 2.29·15-s + 7.38·17-s + 4.58i·19-s − 2.29i·21-s + 2.51·23-s − 25-s + 1.70i·27-s − 4.80i·29-s − 3.09·31-s + ⋯
L(s)  = 1  + 1.32i·3-s − 0.447i·5-s − 0.377·7-s − 0.751·9-s + 0.224i·11-s + 1.06i·13-s + 0.591·15-s + 1.79·17-s + 1.05i·19-s − 0.500i·21-s + 0.523·23-s − 0.200·25-s + 0.328i·27-s − 0.891i·29-s − 0.555·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.521963742\)
\(L(\frac12)\) \(\approx\) \(1.521963742\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + T \)
good3 \( 1 - 2.29iT - 3T^{2} \)
11 \( 1 - 0.744iT - 11T^{2} \)
13 \( 1 - 3.83iT - 13T^{2} \)
17 \( 1 - 7.38T + 17T^{2} \)
19 \( 1 - 4.58iT - 19T^{2} \)
23 \( 1 - 2.51T + 23T^{2} \)
29 \( 1 + 4.80iT - 29T^{2} \)
31 \( 1 + 3.09T + 31T^{2} \)
37 \( 1 + 11.6iT - 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 1.83T + 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 - 11.0iT - 59T^{2} \)
61 \( 1 - 8.51iT - 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 - 3.41T + 71T^{2} \)
73 \( 1 + 8.51T + 73T^{2} \)
79 \( 1 + 8.36T + 79T^{2} \)
83 \( 1 - 4.58iT - 83T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 + 3.38T + 97T^{2} \)
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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

−9.323489999163891306412226953544, −8.920425685634941199064089835107, −7.82141725941560861789743849305, −7.10957768405317831640662136473, −5.79834433809768748447374985591, −5.44457580676116779578155352011, −4.19014763308781626847662267030, −3.97812349003314518335649740095, −2.83285097433587061983130582085, −1.37355136144138289882015142686, 0.56563651343461645109214901649, 1.61587753810156607365718105453, 2.93491750803413999228462643034, 3.35969732343852794289586471881, 4.99943899652972576363103865919, 5.73434249684400743738535240624, 6.63878150391994228799180016125, 7.11118850038665922263449866734, 7.907974257995763630127988178205, 8.422737323195057052221147365247

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