Properties

Label 2-2240-140.139-c1-0-4
Degree $2$
Conductor $2240$
Sign $-0.910 - 0.413i$
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  + 0.326i·3-s + (−0.278 − 2.21i)5-s + (2.25 + 1.38i)7-s + 2.89·9-s + 4.91i·11-s − 4.89·13-s + (0.724 − 0.0908i)15-s − 6.14·17-s − 4.72·19-s + (−0.452 + 0.735i)21-s − 6.30·23-s + (−4.84 + 1.23i)25-s + 1.92i·27-s − 4.66·29-s + 4.98·31-s + ⋯
L(s)  = 1  + 0.188i·3-s + (−0.124 − 0.992i)5-s + (0.852 + 0.523i)7-s + 0.964·9-s + 1.48i·11-s − 1.35·13-s + (0.187 − 0.0234i)15-s − 1.49·17-s − 1.08·19-s + (−0.0986 + 0.160i)21-s − 1.31·23-s + (−0.969 + 0.247i)25-s + 0.370i·27-s − 0.866·29-s + 0.895·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.910 - 0.413i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.910 - 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4514116206\)
\(L(\frac12)\) \(\approx\) \(0.4514116206\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.278 + 2.21i)T \)
7 \( 1 + (-2.25 - 1.38i)T \)
good3 \( 1 - 0.326iT - 3T^{2} \)
11 \( 1 - 4.91iT - 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 + 6.14T + 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
23 \( 1 + 6.30T + 23T^{2} \)
29 \( 1 + 4.66T + 29T^{2} \)
31 \( 1 - 4.98T + 31T^{2} \)
37 \( 1 + 2.88iT - 37T^{2} \)
41 \( 1 - 7.38iT - 41T^{2} \)
43 \( 1 + 6.20T + 43T^{2} \)
47 \( 1 + 5.09iT - 47T^{2} \)
53 \( 1 - 1.63iT - 53T^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 + 8.50iT - 61T^{2} \)
67 \( 1 - 3.67T + 67T^{2} \)
71 \( 1 + 1.27iT - 71T^{2} \)
73 \( 1 + 8.00T + 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + 0.484iT - 83T^{2} \)
89 \( 1 + 11.3iT - 89T^{2} \)
97 \( 1 + 6.14T + 97T^{2} \)
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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

−9.452549793559519631936367831445, −8.601002457178513553580756553757, −7.85787571428098725062440343017, −7.19785261868697126034036634275, −6.26703182653664746871751053085, −5.00385049020217991997266537503, −4.61852454256312524391217772793, −4.13435213810422371072301448773, −2.14625273788710592249689116543, −1.84837069487193042953933711896, 0.13947791435152056720118461252, 1.82672796515004846745653082650, 2.64400857860347485221349661736, 3.95829550192243277925996902166, 4.43316888281825155316692414918, 5.62672179321194043378354219357, 6.60835970588018991975656572604, 7.05090721534935662630682914405, 7.942618214304178295769948931666, 8.448915141422513527679771114189

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