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Properties

Label 2-2240-16.5-c1-0-38
Degree $2$
Conductor $2240$
Sign $0.0897 + 0.995i$
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  + (1.78 − 1.78i)3-s + (−0.707 − 0.707i)5-s − i·7-s − 3.40i·9-s + (3.20 + 3.20i)11-s + (4.01 − 4.01i)13-s − 2.53·15-s + 2.63·17-s + (1.42 − 1.42i)19-s + (−1.78 − 1.78i)21-s + 7.25i·23-s + 1.00i·25-s + (−0.725 − 0.725i)27-s + (5.26 − 5.26i)29-s + 3.94·31-s + ⋯
L(s)  = 1  + (1.03 − 1.03i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s − 1.13i·9-s + (0.966 + 0.966i)11-s + (1.11 − 1.11i)13-s − 0.653·15-s + 0.640·17-s + (0.326 − 0.326i)19-s + (−0.390 − 0.390i)21-s + 1.51i·23-s + 0.200i·25-s + (−0.139 − 0.139i)27-s + (0.978 − 0.978i)29-s + 0.708·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.0897 + 0.995i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.0897 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.815262075\)
\(L(\frac12)\) \(\approx\) \(2.815262075\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.78i)T - 3iT^{2} \)
11 \( 1 + (-3.20 - 3.20i)T + 11iT^{2} \)
13 \( 1 + (-4.01 + 4.01i)T - 13iT^{2} \)
17 \( 1 - 2.63T + 17T^{2} \)
19 \( 1 + (-1.42 + 1.42i)T - 19iT^{2} \)
23 \( 1 - 7.25iT - 23T^{2} \)
29 \( 1 + (-5.26 + 5.26i)T - 29iT^{2} \)
31 \( 1 - 3.94T + 31T^{2} \)
37 \( 1 + (4.55 + 4.55i)T + 37iT^{2} \)
41 \( 1 - 4.78iT - 41T^{2} \)
43 \( 1 + (3.44 + 3.44i)T + 43iT^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (3.16 + 3.16i)T + 53iT^{2} \)
59 \( 1 + (-5.09 - 5.09i)T + 59iT^{2} \)
61 \( 1 + (9.21 - 9.21i)T - 61iT^{2} \)
67 \( 1 + (-5.47 + 5.47i)T - 67iT^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + 4.28iT - 73T^{2} \)
79 \( 1 + 4.27T + 79T^{2} \)
83 \( 1 + (10.7 - 10.7i)T - 83iT^{2} \)
89 \( 1 + 0.251iT - 89T^{2} \)
97 \( 1 + 1.49T + 97T^{2} \)
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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

−8.608129959948751320315610881534, −8.047299448999686900032944852923, −7.46135200060625148523019262547, −6.77581927300742071772066175707, −5.88059729762512832499376117536, −4.74205454939787557735796527253, −3.66174664880008193266973590441, −3.07054134286129646211658428442, −1.71350639377938130072265183227, −1.01773096334295798844862574121, 1.38831645622982359491866823500, 2.84631008042896503172479842006, 3.46699268413729851767420245641, 4.12711862378913389282533164206, 4.99419671876597607226916065177, 6.29551851956855389671801817175, 6.69857908259191484295651476322, 8.143125575083473252255615237681, 8.577309482077511880238385216211, 9.010692282882340263832383541953

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