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Properties

Label 2-2240-140.79-c0-0-2
Degree $2$
Conductor $2240$
Sign $0.832 - 0.553i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 0.999·15-s + (−0.499 + 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + 27-s + 29-s + 0.999·35-s − 41-s − 43-s + (1 − 1.73i)47-s + (−0.499 + 0.866i)49-s + (−0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 0.999·15-s + (−0.499 + 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + 27-s + 29-s + 0.999·35-s − 41-s − 43-s + (1 − 1.73i)47-s + (−0.499 + 0.866i)49-s + (−0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.641951531\)
\(L(\frac12)\) \(\approx\) \(1.641951531\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.223406622793722033889358104758, −8.620503445444294818241416391040, −8.192972544249787449631457148575, −6.96076898347597237443055853984, −5.91489349262696394382811965517, −5.22124208985658920448227472268, −4.54091209532366868866437900357, −3.64564293790094602755200480004, −2.55750065147470240143626632587, −1.48584274320279708404662926580, 1.34240989926389993025326318431, 2.26447322081202494696376767729, 3.12998924668221957986943907047, 4.25259763376587021656123422318, 5.18398105297171602371041154256, 6.46857377785885349065098926576, 6.74158016808290239950535409007, 7.71507901523219204175097940948, 8.060679129234580297922630404262, 9.052166789529282406851037071577

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