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Properties

Label 2-2340-39.2-c1-0-5
Degree $2$
Conductor $2340$
Sign $0.991 - 0.128i$
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 + (−4.01 + 1.07i)7-s + (−3.52 − 0.945i)11-s + (−3.53 − 0.711i)13-s + (−0.250 + 0.433i)17-s + (−0.655 − 2.44i)19-s + (4.31 + 7.47i)23-s + 1.00i·25-s + (1.88 − 1.09i)29-s + (7.27 − 7.27i)31-s + (3.59 + 2.07i)35-s + (−0.373 + 1.39i)37-s + (−0.600 + 2.23i)41-s + (10.4 + 6.04i)43-s + (3.63 − 3.63i)47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + (−1.51 + 0.406i)7-s + (−1.06 − 0.285i)11-s + (−0.980 − 0.197i)13-s + (−0.0607 + 0.105i)17-s + (−0.150 − 0.561i)19-s + (0.899 + 1.55i)23-s + 0.200i·25-s + (0.350 − 0.202i)29-s + (1.30 − 1.30i)31-s + (0.607 + 0.350i)35-s + (−0.0614 + 0.229i)37-s + (−0.0937 + 0.349i)41-s + (1.59 + 0.921i)43-s + (0.530 − 0.530i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9122208255\)
\(L(\frac12)\) \(\approx\) \(0.9122208255\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.53 + 0.711i)T \)
good7 \( 1 + (4.01 - 1.07i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.52 + 0.945i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.250 - 0.433i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.655 + 2.44i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.31 - 7.47i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.88 + 1.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.27 + 7.27i)T - 31iT^{2} \)
37 \( 1 + (0.373 - 1.39i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.600 - 2.23i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.4 - 6.04i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.63 + 3.63i)T - 47iT^{2} \)
53 \( 1 + 3.27iT - 53T^{2} \)
59 \( 1 + (-2.14 - 8.01i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.60 + 7.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.6 - 3.40i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (12.8 - 3.43i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.04 + 7.04i)T + 73iT^{2} \)
79 \( 1 - 2.00T + 79T^{2} \)
83 \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \)
89 \( 1 + (12.3 + 3.30i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.79 + 10.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.175129850294516661828488091762, −8.181143628400985401404043933666, −7.49742809041173059983355133933, −6.71000046562320101325485470699, −5.81412983697735202656407622775, −5.15718082082328014655521132990, −4.14225927732840622798418937294, −3.01816956767775353556839938701, −2.53767416122143166420473553980, −0.62279452538100313184209586464, 0.54239951755084797096342229799, 2.51387035658936927695133346592, 2.99855326051554881366080758461, 4.12027462127258784888317016797, 4.92496264308533021634545578938, 5.96005165512858904971492583748, 6.89517569427798091807130170830, 7.16025766870315189593978736835, 8.188059007162405136155297261221, 9.007884733705521267690239154012

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