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Properties

Label 2-2340-13.3-c1-0-18
Degree $2$
Conductor $2340$
Sign $-0.872 + 0.488i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + (−1.5 + 2.59i)7-s + (1.5 + 2.59i)11-s + (1 − 3.46i)13-s + (−3.5 + 6.06i)17-s + (−0.5 + 0.866i)19-s + (−3.5 − 6.06i)23-s + 25-s + (−2.5 − 4.33i)29-s − 4·31-s + (1.5 − 2.59i)35-s + (1.5 + 2.59i)37-s + (3.5 + 6.06i)41-s + (4.5 − 7.79i)43-s − 8·47-s + ⋯
L(s)  = 1  − 0.447·5-s + (−0.566 + 0.981i)7-s + (0.452 + 0.783i)11-s + (0.277 − 0.960i)13-s + (−0.848 + 1.47i)17-s + (−0.114 + 0.198i)19-s + (−0.729 − 1.26i)23-s + 0.200·25-s + (−0.464 − 0.804i)29-s − 0.718·31-s + (0.253 − 0.439i)35-s + (0.246 + 0.427i)37-s + (0.546 + 0.946i)41-s + (0.686 − 1.18i)43-s − 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (-1 + 3.46i)T \)
good7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.5 - 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.5 + 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.531230751751306782381720014366, −8.129896573527211396526231334783, −7.07333674843593360184764389978, −6.20429103168790069535178946288, −5.75314928937247293095406385081, −4.50173267951935701984822661979, −3.84311694788146426607907911327, −2.74699573493411003776391488007, −1.80747109841177323636992455344, 0, 1.30256469810780121218017699756, 2.75771957710828053636665970441, 3.80261664299451090038676524863, 4.22562390836046947820236975264, 5.40212614971475008094060306723, 6.33642922060703740205905321251, 7.17183870093817072046984036586, 7.46544604244386318895949268125, 8.773453692928977125871023141085

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