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Properties

Label 2-2240-16.5-c1-0-39
Degree $2$
Conductor $2240$
Sign $-0.875 + 0.482i$
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  + (−0.257 + 0.257i)3-s + (−0.707 − 0.707i)5-s − i·7-s + 2.86i·9-s + (1.38 + 1.38i)11-s + (−1.57 + 1.57i)13-s + 0.364·15-s − 5.96·17-s + (1.21 − 1.21i)19-s + (0.257 + 0.257i)21-s − 5.13i·23-s + 1.00i·25-s + (−1.51 − 1.51i)27-s + (2.01 − 2.01i)29-s + 0.281·31-s + ⋯
L(s)  = 1  + (−0.148 + 0.148i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s + 0.955i·9-s + (0.416 + 0.416i)11-s + (−0.437 + 0.437i)13-s + 0.0941·15-s − 1.44·17-s + (0.279 − 0.279i)19-s + (0.0562 + 0.0562i)21-s − 1.07i·23-s + 0.200i·25-s + (−0.291 − 0.291i)27-s + (0.374 − 0.374i)29-s + 0.0505·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.875 + 0.482i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2474029770\)
\(L(\frac12)\) \(\approx\) \(0.2474029770\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (0.257 - 0.257i)T - 3iT^{2} \)
11 \( 1 + (-1.38 - 1.38i)T + 11iT^{2} \)
13 \( 1 + (1.57 - 1.57i)T - 13iT^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 + (-1.21 + 1.21i)T - 19iT^{2} \)
23 \( 1 + 5.13iT - 23T^{2} \)
29 \( 1 + (-2.01 + 2.01i)T - 29iT^{2} \)
31 \( 1 - 0.281T + 31T^{2} \)
37 \( 1 + (2.94 + 2.94i)T + 37iT^{2} \)
41 \( 1 - 7.72iT - 41T^{2} \)
43 \( 1 + (7.82 + 7.82i)T + 43iT^{2} \)
47 \( 1 - 9.99T + 47T^{2} \)
53 \( 1 + (8.16 + 8.16i)T + 53iT^{2} \)
59 \( 1 + (4.52 + 4.52i)T + 59iT^{2} \)
61 \( 1 + (6.03 - 6.03i)T - 61iT^{2} \)
67 \( 1 + (-7.46 + 7.46i)T - 67iT^{2} \)
71 \( 1 + 4.27iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (8.88 - 8.88i)T - 83iT^{2} \)
89 \( 1 + 7.06iT - 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−8.656953022763056981516552325568, −8.020711386886110910320083793872, −7.04848044891698951078677273834, −6.59802014029803256939046389923, −5.34370166522573661268589148044, −4.55824881305045176286048743995, −4.13375764214634271745203230684, −2.69179762016154584212318246857, −1.73755896926402276473531452539, −0.086161430928692759859043372185, 1.40684477793482517406959593712, 2.77236132977611748284421124923, 3.55582545672054206444766287319, 4.49347511113471542601551499148, 5.54575043120029720529863062893, 6.30736045087905411169823102641, 6.96197534153882830109124914549, 7.75141560243506481189522469829, 8.733741993765026996136838039218, 9.209878944304829456079593102344

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