Properties

Label 2-2340-13.12-c1-0-6
Degree $2$
Conductor $2340$
Sign $0.183 - 0.983i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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·5-s + 3.88i·7-s − 1.54i·11-s + (3.54 + 0.660i)13-s + 2.86i·19-s − 5.42·23-s − 25-s + 5.20·29-s + 6.22i·31-s + 3.88·35-s − 8.56i·37-s + 9.08i·41-s + 0.980·43-s + 6.52i·47-s − 8.08·49-s + ⋯
L(s)  = 1  − 0.447i·5-s + 1.46i·7-s − 0.465i·11-s + (0.983 + 0.183i)13-s + 0.657i·19-s − 1.13·23-s − 0.200·25-s + 0.966·29-s + 1.11i·31-s + 0.656·35-s − 1.40i·37-s + 1.41i·41-s + 0.149·43-s + 0.951i·47-s − 1.15·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.183 - 0.983i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.556871059\)
\(L(\frac12)\) \(\approx\) \(1.556871059\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (-3.54 - 0.660i)T \)
good7 \( 1 - 3.88iT - 7T^{2} \)
11 \( 1 + 1.54iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.86iT - 19T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 - 5.20T + 29T^{2} \)
31 \( 1 - 6.22iT - 31T^{2} \)
37 \( 1 + 8.56iT - 37T^{2} \)
41 \( 1 - 9.08iT - 41T^{2} \)
43 \( 1 - 0.980T + 43T^{2} \)
47 \( 1 - 6.52iT - 47T^{2} \)
53 \( 1 + 6.44T + 53T^{2} \)
59 \( 1 + 4.45iT - 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 6.97iT - 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 3.43iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 8.56iT - 83T^{2} \)
89 \( 1 - 17.1iT - 89T^{2} \)
97 \( 1 - 13.7iT - 97T^{2} \)
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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

−9.024323733099214977172254880954, −8.419933562945738606728392776175, −7.961636583332536010065412608558, −6.61051533141047255617364681067, −5.94434041793353008687454180086, −5.40770103827613335041664039804, −4.36266850789710813432278711129, −3.38539307946793023244940068001, −2.37715235556412357354197549621, −1.29353098398290518875738647531, 0.56517850377417058969483218714, 1.84166882085968093845363408931, 3.12920944726107936351752383177, 3.98554237600106455541338146060, 4.60323282406559557783553497683, 5.82318756662032791539946463017, 6.62482424578790494525679453021, 7.22857823775016141467566672477, 7.973856469274578264471913558754, 8.709125234480929070955319704239

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