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Properties

Label 2-2340-39.8-c1-0-6
Degree $2$
Conductor $2340$
Sign $0.928 - 0.370i$
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  + (−0.707 + 0.707i)5-s + (2.39 − 2.39i)7-s + (1.91 + 1.91i)11-s + (2.69 + 2.39i)13-s + 1.97·17-s + (2 + 2i)19-s − 7.21·23-s − 1.00i·25-s − 2.41i·29-s + (4.79 + 4.79i)31-s + 3.39i·35-s + (0.692 − 0.692i)37-s + (−4.86 + 4.86i)41-s − 11.0i·43-s + (7.21 + 7.21i)47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + (0.906 − 0.906i)7-s + (0.576 + 0.576i)11-s + (0.746 + 0.665i)13-s + 0.479·17-s + (0.458 + 0.458i)19-s − 1.50·23-s − 0.200i·25-s − 0.447i·29-s + (0.861 + 0.861i)31-s + 0.573i·35-s + (0.113 − 0.113i)37-s + (−0.760 + 0.760i)41-s − 1.67i·43-s + (1.05 + 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.032334358\)
\(L(\frac12)\) \(\approx\) \(2.032334358\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-2.69 - 2.39i)T \)
good7 \( 1 + (-2.39 + 2.39i)T - 7iT^{2} \)
11 \( 1 + (-1.91 - 1.91i)T + 11iT^{2} \)
17 \( 1 - 1.97T + 17T^{2} \)
19 \( 1 + (-2 - 2i)T + 19iT^{2} \)
23 \( 1 + 7.21T + 23T^{2} \)
29 \( 1 + 2.41iT - 29T^{2} \)
31 \( 1 + (-4.79 - 4.79i)T + 31iT^{2} \)
37 \( 1 + (-0.692 + 0.692i)T - 37iT^{2} \)
41 \( 1 + (4.86 - 4.86i)T - 41iT^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (-7.21 - 7.21i)T + 47iT^{2} \)
53 \( 1 + 0.0188iT - 53T^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + 6.91T + 61T^{2} \)
67 \( 1 + (-3.38 - 3.38i)T + 67iT^{2} \)
71 \( 1 + (-3.90 + 3.90i)T - 71iT^{2} \)
73 \( 1 + (-3.09 + 3.09i)T - 73iT^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + (-0.563 + 0.563i)T - 83iT^{2} \)
89 \( 1 + (-9.11 - 9.11i)T + 89iT^{2} \)
97 \( 1 + (0.103 + 0.103i)T + 97iT^{2} \)
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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

−8.989742076870886054010711288660, −8.062570240014059818472511406944, −7.62868938002279940790228682756, −6.76251923528092118421221758083, −6.05078409948692781714787954951, −4.89168454460203146138780617776, −4.12060735607655627769305462544, −3.54579669657003061357751104719, −2.03980516421781238296375450159, −1.10586543736400547055488718829, 0.871417268226724720676076712630, 2.02336534518584650297919908448, 3.20188513007646359223996106303, 4.08332050397982825584108586448, 5.07975960103753605198948966628, 5.74919833442308218566116623395, 6.45213458679836175599080623733, 7.72433669018580683868024432558, 8.187824682012725258950956741791, 8.789567426371392224531490624910

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