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Properties

Label 2-2340-39.20-c1-0-4
Degree $2$
Conductor $2340$
Sign $0.452 - 0.891i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.141225453\)
\(L(\frac12)\) \(\approx\) \(1.141225453\)
\(L(\frac{3}{2})\) 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

−9.310063286731684117161156244876, −8.477540147452960623341442589824, −7.43849318577985487599831832007, −6.75347099627073041090042072274, −6.09381217070300503185879172697, −5.30929425213071909679877876340, −4.06294230412179538947061800593, −3.54332813343631315660279924066, −2.38374945362484844404406234299, −1.07068883498336570685072083538, 0.43742029508050861327364343731, 2.24505892102176769753447137286, 2.87978061045108591875063240483, 3.95462566114646841303094556630, 4.80736053288755916264579983300, 6.10826158488662630451565382968, 6.35975533397950755105812571538, 7.13127045363040184104869160023, 8.094257249541245268641360253780, 9.103565573342523116220384161006

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