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Properties

Label 2-2240-280.139-c1-0-50
Degree $2$
Conductor $2240$
Sign $0.258 - 0.965i$
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  + 3.40·3-s + 2.23i·5-s + 2.64i·7-s + 8.62·9-s − 5.55·11-s + 1.06i·13-s + 7.62i·15-s + 5.90·17-s + 9.02i·21-s − 5.00·25-s + 19.1·27-s − 4.83i·29-s − 18.9·33-s − 5.91·35-s + 3.62i·39-s + ⋯
L(s)  = 1  + 1.96·3-s + 0.999i·5-s + 0.999i·7-s + 2.87·9-s − 1.67·11-s + 0.294i·13-s + 1.96i·15-s + 1.43·17-s + 1.96i·21-s − 1.00·25-s + 3.68·27-s − 0.898i·29-s − 3.29·33-s − 0.999·35-s + 0.580i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.533040436\)
\(L(\frac12)\) \(\approx\) \(3.533040436\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 3.40T + 3T^{2} \)
11 \( 1 + 5.55T + 11T^{2} \)
13 \( 1 - 1.06iT - 13T^{2} \)
17 \( 1 - 5.90T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4.83iT - 29T^{2} \)
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 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 14.8iT - 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 3.45T + 97T^{2} \)
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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

−9.229701665975885603141259179236, −8.141507026410530632920954230048, −7.936856185171942173455610631866, −7.24294499887491127310329141937, −6.18091425534586697788935751508, −5.16681691654144022406272709791, −4.01729215646274164664257436968, −2.94394644002773153138458507263, −2.77024271382166652487278440272, −1.82013965926510509209270971170, 0.962479771436886900368833346641, 2.04701533977222520747631009638, 3.12532462279458830867612995015, 3.74183575612223304596275866565, 4.71551265291906984980448732202, 5.43023622994733871810924916970, 7.03531498754480118543257652090, 7.65308248631192491051698651774, 8.127386568342078406471689430966, 8.658142979091596430058185375018

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