Properties

Label 2-2240-140.139-c1-0-14
Degree $2$
Conductor $2240$
Sign $-i$
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  − 0.613i·3-s − 2.23·5-s − 2.64i·7-s + 2.62·9-s + 5.55i·11-s + 1.06·13-s + 1.37i·15-s − 5.75·17-s − 1.62·21-s + 5.00·25-s − 3.45i·27-s − 9.62·29-s + 3.40·33-s + 5.91i·35-s − 0.652i·39-s + ⋯
L(s)  = 1  − 0.354i·3-s − 0.999·5-s − 0.999i·7-s + 0.874·9-s + 1.67i·11-s + 0.294·13-s + 0.354i·15-s − 1.39·17-s − 0.354·21-s + 1.00·25-s − 0.664i·27-s − 1.78·29-s + 0.593·33-s + 0.999i·35-s − 0.104i·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7786642098\)
\(L(\frac12)\) \(\approx\) \(0.7786642098\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 0.613iT - 3T^{2} \)
11 \( 1 - 5.55iT - 11T^{2} \)
13 \( 1 - 1.06T + 13T^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9.62T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 13.6iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 - 9.74iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 19.3T + 97T^{2} \)
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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

−9.352772732985893543405551440115, −8.219922751496078916983212929702, −7.47925380406841560858540693747, −7.10534922443687349148249311043, −6.48580221811321164835965682510, −4.96027897523411492194083412884, −4.24049068364168777709788665729, −3.83927517456956992741399837900, −2.30055851495701987458672385697, −1.23484645323034467890196917071, 0.28784622818536308819919780910, 1.93497822160919932889858279781, 3.23518706687415579094348172684, 3.83857834770084983194978225423, 4.78024269885192141624202915714, 5.65329805451712424615338879573, 6.49199302696941049004786368363, 7.33773644622475018274537531047, 8.266964275535183611635011241872, 8.804522171571868093702367257149

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