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Properties

Label 2-2240-28.27-c1-0-15
Degree $2$
Conductor $2240$
Sign $-0.313 - 0.949i$
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  + 1.77·3-s + i·5-s + (−0.829 − 2.51i)7-s + 0.148·9-s + 6.36i·11-s + 0.563i·13-s + 1.77i·15-s + 6.12i·17-s − 0.694·19-s + (−1.47 − 4.45i)21-s + 0.964i·23-s − 25-s − 5.05·27-s + 2.02·29-s − 8.09·31-s + ⋯
L(s)  = 1  + 1.02·3-s + 0.447i·5-s + (−0.313 − 0.949i)7-s + 0.0494·9-s + 1.91i·11-s + 0.156i·13-s + 0.458i·15-s + 1.48i·17-s − 0.159·19-s + (−0.321 − 0.972i)21-s + 0.201i·23-s − 0.200·25-s − 0.973·27-s + 0.376·29-s − 1.45·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.666074265\)
\(L(\frac12)\) \(\approx\) \(1.666074265\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (0.829 + 2.51i)T \)
good3 \( 1 - 1.77T + 3T^{2} \)
11 \( 1 - 6.36iT - 11T^{2} \)
13 \( 1 - 0.563iT - 13T^{2} \)
17 \( 1 - 6.12iT - 17T^{2} \)
19 \( 1 + 0.694T + 19T^{2} \)
23 \( 1 - 0.964iT - 23T^{2} \)
29 \( 1 - 2.02T + 29T^{2} \)
31 \( 1 + 8.09T + 31T^{2} \)
37 \( 1 + 1.89T + 37T^{2} \)
41 \( 1 + 1.97iT - 41T^{2} \)
43 \( 1 + 5.26iT - 43T^{2} \)
47 \( 1 + 3.03T + 47T^{2} \)
53 \( 1 + 1.73T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 9.96iT - 61T^{2} \)
67 \( 1 - 0.714iT - 67T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + 0.743iT - 73T^{2} \)
79 \( 1 - 3.94iT - 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 16.6iT - 89T^{2} \)
97 \( 1 - 9.54iT - 97T^{2} \)
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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

−9.273505935424915108507677100587, −8.486362656894184975709087952870, −7.62019835715448990265447626030, −7.15946048832887129550014592040, −6.39986864858527720378111583940, −5.21666709758801378717598616590, −4.00717119193341683033486905201, −3.72031722675938193164697186938, −2.43852186447621137378008543272, −1.68922583224964391536238863893, 0.46456606692977132297619422273, 2.09389519222259167991622231345, 3.07732222509807993187717642623, 3.44667100712521279029086503311, 4.89519631398334104801575825776, 5.65316541134470301136772107839, 6.33909096443161916817264185871, 7.51402260535122635545886482376, 8.294845789752969052932622395865, 8.829367785190630343047071406419

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