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Properties

Label 2-2240-35.34-c0-0-5
Degree $2$
Conductor $2240$
Sign $-0.707 + 0.707i$
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  − 1.41·3-s + i·5-s + (−0.707 − 0.707i)7-s + 1.00·9-s − 1.41i·15-s + (1.00 + 1.00i)21-s + 1.41i·23-s − 25-s − 2·29-s + (0.707 − 0.707i)35-s − 2i·41-s − 1.41i·43-s + 1.00i·45-s − 1.41·47-s + 1.00i·49-s + ⋯
L(s)  = 1  − 1.41·3-s + i·5-s + (−0.707 − 0.707i)7-s + 1.00·9-s − 1.41i·15-s + (1.00 + 1.00i)21-s + 1.41i·23-s − 25-s − 2·29-s + (0.707 − 0.707i)35-s − 2i·41-s − 1.41i·43-s + 1.00i·45-s − 1.41·47-s + 1.00i·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1243214858\)
\(L(\frac12)\) \(\approx\) \(0.1243214858\)
\(L(1)\) 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 \( 1 - iT \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.250983460806488235020417980558, −7.80629157293174856835116057324, −7.13226119630741757628800288830, −6.62931597850986804223674062900, −5.79656272267362009738392787127, −5.24609300655022092760602989252, −3.93054745137848606669445419162, −3.35818709159905837849051285624, −1.84924287473662587634657924794, −0.10584117177015804727069645033, 1.36494030425422505290753424538, 2.78734616764231358974668198617, 4.15552479623073398087283616812, 4.89356266618034015865649900991, 5.66869594877539609422565847783, 6.14641907453021981221506252895, 6.93521515225505121331750837474, 8.056175574626407129507678599862, 8.801582010302082528183075535201, 9.604920883842881768658789892852

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