Properties

Label 2-2240-56.27-c1-0-30
Degree $2$
Conductor $2240$
Sign $0.194 + 0.980i$
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  − 0.237i·3-s − 5-s + (−2.63 − 0.174i)7-s + 2.94·9-s − 4.69·11-s + 2.77·13-s + 0.237i·15-s + 3.55i·17-s + 6.72i·19-s + (−0.0415 + 0.627i)21-s − 8.45i·23-s + 25-s − 1.41i·27-s − 5.80i·29-s + 6.92·31-s + ⋯
L(s)  = 1  − 0.137i·3-s − 0.447·5-s + (−0.997 − 0.0660i)7-s + 0.981·9-s − 1.41·11-s + 0.770·13-s + 0.0613i·15-s + 0.862i·17-s + 1.54i·19-s + (−0.00906 + 0.136i)21-s − 1.76i·23-s + 0.200·25-s − 0.271i·27-s − 1.07i·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.072743851\)
\(L(\frac12)\) \(\approx\) \(1.072743851\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (2.63 + 0.174i)T \)
good3 \( 1 + 0.237iT - 3T^{2} \)
11 \( 1 + 4.69T + 11T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
17 \( 1 - 3.55iT - 17T^{2} \)
19 \( 1 - 6.72iT - 19T^{2} \)
23 \( 1 + 8.45iT - 23T^{2} \)
29 \( 1 + 5.80iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 0.171iT - 37T^{2} \)
41 \( 1 + 5.71iT - 41T^{2} \)
43 \( 1 - 3.26T + 43T^{2} \)
47 \( 1 + 2.83T + 47T^{2} \)
53 \( 1 + 7.09iT - 53T^{2} \)
59 \( 1 + 4.79iT - 59T^{2} \)
61 \( 1 + 9.63T + 61T^{2} \)
67 \( 1 + 9.99T + 67T^{2} \)
71 \( 1 + 7.49iT - 71T^{2} \)
73 \( 1 + 8.35iT - 73T^{2} \)
79 \( 1 + 5.05iT - 79T^{2} \)
83 \( 1 + 9.53iT - 83T^{2} \)
89 \( 1 - 5.75iT - 89T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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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

−8.693990276867878172365358790834, −8.038300741787117346732886261954, −7.46043557859814991304030324066, −6.34218904983245919312686174725, −6.03865938167241495764015681035, −4.69745561496685040246683151745, −3.96073840767427174802098294484, −3.09488313088641524034293843478, −1.95093141604055007549877308202, −0.43887923615912274169739186707, 1.04626951358035518279679211983, 2.70839701425074418929263510939, 3.30529678915919397135456953905, 4.42222486072448536658126493512, 5.14250890694731643382620682220, 6.09633862561736054573813288554, 7.12929295686348244359213115614, 7.41646983828087260795047407856, 8.473582568495222443243972189239, 9.323539825085718043147209935520

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