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Properties

Label 2-2240-140.39-c0-0-3
Degree $2$
Conductor $2240$
Sign $-0.553 + 0.832i$
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  + (0.866 − 1.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + (−1 − 1.73i)9-s − 1.73·15-s + (1.5 − 0.866i)21-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s − 1.73·27-s + 29-s − 0.999i·35-s + 41-s − 1.73·43-s + (−1 + 1.73i)45-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + (−1 − 1.73i)9-s − 1.73·15-s + (1.5 − 0.866i)21-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s − 1.73·27-s + 29-s − 0.999i·35-s + 41-s − 1.73·43-s + (−1 + 1.73i)45-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.508222108\)
\(L(\frac12)\) \(\approx\) \(1.508222108\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
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

−8.534669255616383657597846182557, −8.284711421416115061303373179080, −7.66028082875664684628393081876, −6.79583859394006231018577413583, −5.97748033828929447693342576953, −4.94811284309135931179561402270, −4.05595701494413920159529219102, −2.79297657836811247822203636972, −1.97403236975498787247187307384, −0.995465063009623840943227729977, 2.00700563404280842682165738803, 3.15086708779710540534563318787, 3.73590462632019633247008068801, 4.48614914201934789342961272042, 5.21017691248639200814095168590, 6.38373625051062535054015521722, 7.54539077335710462413053001828, 7.968128670633365355090291440202, 8.712265794360440474014174748398, 9.622397850171541925655973331182

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