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Properties

Label 2-2240-280.139-c1-0-79
Degree $2$
Conductor $2240$
Sign $0.874 + 0.485i$
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  + 2.97·3-s + (2.14 − 0.615i)5-s − 2.64·7-s + 5.82·9-s − 7.17i·13-s + (6.38 − 1.82i)15-s + 6.81i·19-s − 7.86·21-s + 7.48·23-s + (4.24 − 2.64i)25-s + 8.40·27-s + (−5.68 + 1.62i)35-s − 21.3i·39-s + (12.5 − 3.58i)45-s + 7.00·49-s + ⋯
L(s)  = 1  + 1.71·3-s + (0.961 − 0.275i)5-s − 0.999·7-s + 1.94·9-s − 1.98i·13-s + (1.64 − 0.472i)15-s + 1.56i·19-s − 1.71·21-s + 1.56·23-s + (0.848 − 0.529i)25-s + 1.61·27-s + (−0.961 + 0.275i)35-s − 3.41i·39-s + (1.86 − 0.534i)45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.874 + 0.485i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.705564414\)
\(L(\frac12)\) \(\approx\) \(3.705564414\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 + 0.615i)T \)
7 \( 1 + 2.64T \)
good3 \( 1 - 2.97T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7.17iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 6.81iT - 19T^{2} \)
23 \( 1 - 7.48T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 13.9iT - 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 16.9iT - 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.070004048369398968962880443114, −8.220583518337467473200989052497, −7.72602731683279545616830707824, −6.72431189860753129834712308899, −5.85199271020376072015731510868, −5.00859981863712613965187062171, −3.62246841741757939642282501724, −3.12679183478438984382278897589, −2.35531613761783192972019911292, −1.13667687268288245940607972511, 1.50380259248298829023981200663, 2.53601852510950365070830618253, 2.98199486544879608708490192100, 4.04169706087552272628887780158, 4.91182608609113181283114900133, 6.30280892437163395686063635816, 6.92098554878271861157148904043, 7.37609025356044036254891142475, 8.805677230532636916622966101775, 9.133691262431822833169703762920

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