Properties

Label 2-2640-5.4-c1-0-6
Degree $2$
Conductor $2640$
Sign $-0.990 - 0.139i$
Analytic cond. $21.0805$
Root an. cond. $4.59135$
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  + i·3-s + (0.311 − 2.21i)5-s + 4.90i·7-s − 9-s + 11-s + 4.14i·13-s + (2.21 + 0.311i)15-s + 5.33i·17-s − 5.18·19-s − 4.90·21-s − 4i·23-s + (−4.80 − 1.37i)25-s i·27-s − 1.80·29-s − 2.62·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.139 − 0.990i)5-s + 1.85i·7-s − 0.333·9-s + 0.301·11-s + 1.15i·13-s + (0.571 + 0.0803i)15-s + 1.29i·17-s − 1.18·19-s − 1.06·21-s − 0.834i·23-s + (−0.961 − 0.275i)25-s − 0.192i·27-s − 0.335·29-s − 0.470·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.990 - 0.139i$
Analytic conductor: \(21.0805\)
Root analytic conductor: \(4.59135\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2640} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2640,\ (\ :1/2),\ -0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8813955548\)
\(L(\frac12)\) \(\approx\) \(0.8813955548\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (-0.311 + 2.21i)T \)
11 \( 1 - T \)
good7 \( 1 - 4.90iT - 7T^{2} \)
13 \( 1 - 4.14iT - 13T^{2} \)
17 \( 1 - 5.33iT - 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 1.80T + 29T^{2} \)
31 \( 1 + 2.62T + 31T^{2} \)
37 \( 1 + 5.80iT - 37T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 + 4.90iT - 43T^{2} \)
47 \( 1 - 7.05iT - 47T^{2} \)
53 \( 1 + 7.18iT - 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 - 0.755T + 61T^{2} \)
67 \( 1 + 4.85iT - 67T^{2} \)
71 \( 1 + 0.428T + 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 6.42T + 79T^{2} \)
83 \( 1 - 2.90iT - 83T^{2} \)
89 \( 1 + 0.622T + 89T^{2} \)
97 \( 1 - 2.75iT - 97T^{2} \)
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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

−9.139802904578349493196110997163, −8.600327972004237773635333529809, −8.186505105761158373685278597371, −6.67463345187058765259299951920, −5.96322073996326127071579241637, −5.40661109872207471851989109025, −4.46594944128980453040777266346, −3.86091782906617814168235339385, −2.39371914054203298518144493601, −1.77532473541198652605608151932, 0.27579647668228536578355633447, 1.46287185118712345549711804443, 2.77514270505209182235749349670, 3.52303173437102395124376152919, 4.39892290537778238486795716724, 5.51618257470267948394476853436, 6.45878661838847249585147021706, 7.07104601831783766970802977765, 7.53628579320893297039126451593, 8.190603013709827337523653058503

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