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Properties

Label 2-2340-12.11-c1-0-31
Degree $2$
Conductor $2340$
Sign $-0.974 - 0.223i$
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  + (0.273 + 1.38i)2-s + (−1.84 + 0.760i)4-s + i·5-s + 1.96i·7-s + (−1.56 − 2.35i)8-s + (−1.38 + 0.273i)10-s + 4.36·11-s + 13-s + (−2.72 + 0.538i)14-s + (2.84 − 2.81i)16-s + 0.0716i·17-s + 5.97i·19-s + (−0.760 − 1.84i)20-s + (1.19 + 6.06i)22-s + 2.51·23-s + ⋯
L(s)  = 1  + (0.193 + 0.981i)2-s + (−0.924 + 0.380i)4-s + 0.447i·5-s + 0.743i·7-s + (−0.552 − 0.833i)8-s + (−0.438 + 0.0866i)10-s + 1.31·11-s + 0.277·13-s + (−0.729 + 0.143i)14-s + (0.711 − 0.703i)16-s + 0.0173i·17-s + 1.37i·19-s + (−0.169 − 0.413i)20-s + (0.255 + 1.29i)22-s + 0.524·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.974 - 0.223i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.974 - 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.733747234\)
\(L(\frac12)\) \(\approx\) \(1.733747234\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.273 - 1.38i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 - T \)
good7 \( 1 - 1.96iT - 7T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
17 \( 1 - 0.0716iT - 17T^{2} \)
19 \( 1 - 5.97iT - 19T^{2} \)
23 \( 1 - 2.51T + 23T^{2} \)
29 \( 1 - 8.97iT - 29T^{2} \)
31 \( 1 + 8.59iT - 31T^{2} \)
37 \( 1 - 7.61T + 37T^{2} \)
41 \( 1 + 0.210iT - 41T^{2} \)
43 \( 1 - 3.48iT - 43T^{2} \)
47 \( 1 + 6.45T + 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 + 4.78T + 59T^{2} \)
61 \( 1 + 1.86T + 61T^{2} \)
67 \( 1 + 1.41iT - 67T^{2} \)
71 \( 1 + 5.02T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 + 8.61iT - 79T^{2} \)
83 \( 1 + 5.88T + 83T^{2} \)
89 \( 1 - 6.62iT - 89T^{2} \)
97 \( 1 + 9.32T + 97T^{2} \)
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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

−9.221265632216527066239677305853, −8.515828511753416198873304338925, −7.72668378234930095506414996092, −6.96653921261277741739098307910, −6.08762993805565240376725826888, −5.81335373405086884713849933258, −4.60278228712723945394813937563, −3.81821799063174020921075172238, −2.94164506301598888818190129307, −1.39684210748964771710217795269, 0.62912053563020295790010611172, 1.50796469407269137225675604415, 2.76880734928583528619452661008, 3.79818592485598300600238207243, 4.39911726094749577397455676074, 5.16400610148206085386070540515, 6.26727097538203201272277940523, 7.00004922013096988237502167580, 8.123170552021033714910970738335, 8.879525818138968493582508997626

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