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Properties

Label 2-2240-16.5-c1-0-46
Degree $2$
Conductor $2240$
Sign $-0.917 - 0.396i$
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  + (1.16 − 1.16i)3-s + (0.707 + 0.707i)5-s − i·7-s + 0.295i·9-s + (−3.97 − 3.97i)11-s + (−1.48 + 1.48i)13-s + 1.64·15-s − 6.33·17-s + (−4.80 + 4.80i)19-s + (−1.16 − 1.16i)21-s − 4.65i·23-s + 1.00i·25-s + (3.83 + 3.83i)27-s + (−4.24 + 4.24i)29-s + 0.0294·31-s + ⋯
L(s)  = 1  + (0.671 − 0.671i)3-s + (0.316 + 0.316i)5-s − 0.377i·7-s + 0.0986i·9-s + (−1.19 − 1.19i)11-s + (−0.413 + 0.413i)13-s + 0.424·15-s − 1.53·17-s + (−1.10 + 1.10i)19-s + (−0.253 − 0.253i)21-s − 0.971i·23-s + 0.200i·25-s + (0.737 + 0.737i)27-s + (−0.788 + 0.788i)29-s + 0.00528·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.917 - 0.396i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.917 - 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1251200823\)
\(L(\frac12)\) \(\approx\) \(0.1251200823\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 + 1.16i)T - 3iT^{2} \)
11 \( 1 + (3.97 + 3.97i)T + 11iT^{2} \)
13 \( 1 + (1.48 - 1.48i)T - 13iT^{2} \)
17 \( 1 + 6.33T + 17T^{2} \)
19 \( 1 + (4.80 - 4.80i)T - 19iT^{2} \)
23 \( 1 + 4.65iT - 23T^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 - 0.0294T + 31T^{2} \)
37 \( 1 + (5.73 + 5.73i)T + 37iT^{2} \)
41 \( 1 - 0.919iT - 41T^{2} \)
43 \( 1 + (-3.37 - 3.37i)T + 43iT^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + (-0.0243 - 0.0243i)T + 53iT^{2} \)
59 \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \)
61 \( 1 + (-1.34 + 1.34i)T - 61iT^{2} \)
67 \( 1 + (-9.68 + 9.68i)T - 67iT^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 4.67T + 79T^{2} \)
83 \( 1 + (4.38 - 4.38i)T - 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 - 0.578T + 97T^{2} \)
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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

−8.422692978224331036388360784729, −7.987284840202161425041886270295, −7.04001043320057282140668629109, −6.46396655305749949760103103339, −5.49974412788073706888011894735, −4.54334745487182788985256042595, −3.43564411148583608097371955296, −2.46743326523057562690365818254, −1.84327894861206952470194642060, −0.03387071555318085039589911775, 2.10380393396256256030760963692, 2.60719805334695651997559707442, 3.83888746169175358330485437047, 4.73804696840761844665273338764, 5.21040604142697729390038719192, 6.43033021882562051536342206676, 7.17065285467584421653666314025, 8.183592473321137734456289315867, 8.748185556618547182046477756989, 9.533255288969209864930143277655

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