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Properties

Label 2-2340-13.4-c1-0-5
Degree $2$
Conductor $2340$
Sign $0.566 - 0.823i$
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 + (−1.81 + 1.04i)7-s + (−1.5 − 0.866i)11-s + (−3.59 − 0.331i)13-s + (1.81 + 3.14i)17-s + (0.926 − 0.534i)19-s + (3.90 − 6.77i)23-s − 25-s + (−0.263 + 0.456i)29-s + 5.84i·31-s + (1.04 + 1.81i)35-s + (8.44 + 4.87i)37-s + (3.69 + 2.13i)41-s + (4.67 + 8.09i)43-s + 3.46i·47-s + ⋯
L(s)  = 1  − 0.447i·5-s + (−0.685 + 0.395i)7-s + (−0.452 − 0.261i)11-s + (−0.995 − 0.0918i)13-s + (0.439 + 0.762i)17-s + (0.212 − 0.122i)19-s + (0.815 − 1.41i)23-s − 0.200·25-s + (−0.0489 + 0.0847i)29-s + 1.04i·31-s + (0.177 + 0.306i)35-s + (1.38 + 0.801i)37-s + (0.577 + 0.333i)41-s + (0.712 + 1.23i)43-s + 0.505i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.566 - 0.823i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.566 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.212653778\)
\(L(\frac12)\) \(\approx\) \(1.212653778\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 0.331i)T \)
good7 \( 1 + (1.81 - 1.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.81 - 3.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.926 + 0.534i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.90 + 6.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.263 - 0.456i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + (-8.44 - 4.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.69 - 2.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.67 - 8.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + (-1.21 + 0.701i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.55 - 9.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.38 + 5.41i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.2 + 7.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.64iT - 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 + (-4.78 - 2.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.1 - 7.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.254803042887495160232257825144, −8.257034539448475017545480283933, −7.75360338822753142583835941185, −6.65049070607598746037994673477, −6.05806657891574735566731378342, −5.07969513281439639502965040656, −4.45940854002470733954914748881, −3.15968203332135762830888872964, −2.52465724382750437700388593095, −1.00808187269343514342939983688, 0.48972243828317617337283311988, 2.16076627489012176300770402733, 3.04752457817462912955352447715, 3.88964091627014337845880700973, 4.97154159533787929789383674064, 5.70252886098175221092762178999, 6.66794699279615044805067553443, 7.53174202304609996305886636177, 7.66013953765258876737855019051, 9.153454436276782973011502780888

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