Properties

Label 2-2240-35.34-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(1.975597385\)
\(L(\frac12)\) \(\approx\) \(1.975597385\)
\(L(1)\) 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

−9.125909770087743401389697383895, −8.607460566491536003778470449083, −7.78384591556370904443434960001, −7.31424431941453007342579611298, −6.28011466778037113510506983890, −5.40423386802856022619193045079, −4.20407433506505993895151178118, −3.42050594435959414388165862574, −2.47430116350105595534480659430, −1.99226898323628789385560294789, 1.34666811078089520755033665187, 2.17493756776900941870296590072, 3.51556376065953967518053034108, 4.05178209043252604440803154498, 5.01204313542183171913037913994, 5.83920138823794587695111952648, 7.32444558415404015246482034962, 7.66679280756963260765118705011, 8.359376513667225449201013032334, 9.127467650661190345456698317869

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