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Properties

Label 2-2340-13.9-c1-0-16
Degree $2$
Conductor $2340$
Sign $0.0128 + 0.999i$
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  + 5-s + (−1.5 − 2.59i)7-s + (3 − 5.19i)11-s + (−2.5 + 2.59i)13-s + (3 + 5.19i)17-s + (1 − 1.73i)23-s + 25-s + (4 − 6.92i)29-s + 3·31-s + (−1.5 − 2.59i)35-s + (−3 + 5.19i)37-s + (1 − 1.73i)41-s + (−5.5 − 9.52i)43-s − 12·47-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + 0.447·5-s + (−0.566 − 0.981i)7-s + (0.904 − 1.56i)11-s + (−0.693 + 0.720i)13-s + (0.727 + 1.26i)17-s + (0.208 − 0.361i)23-s + 0.200·25-s + (0.742 − 1.28i)29-s + 0.538·31-s + (−0.253 − 0.439i)35-s + (−0.493 + 0.854i)37-s + (0.156 − 0.270i)41-s + (−0.838 − 1.45i)43-s − 1.75·47-s + (−0.142 + 0.247i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.639614908\)
\(L(\frac12)\) \(\approx\) \(1.639614908\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (2.5 - 2.59i)T \)
good7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4 + 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + 15T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.696437010141285736332475498564, −8.207359073457134040592799359705, −7.07750475110266814187753928377, −6.42760776716733226041916455717, −5.90994803859076546520009083673, −4.72274050678034133638838704896, −3.79989099503595025740514492010, −3.16679318095859523051258697062, −1.75325302574871283423453290592, −0.58480587030308368234404322142, 1.35122029720640795778669461058, 2.53656177201499154078334134807, 3.19094196149999124688083490174, 4.59480016405058275990728969613, 5.20176581313010257890585847625, 6.03722422075810834426544526817, 6.97258341629976125252397841656, 7.41224684048805989029681861675, 8.608144310547725187509109870015, 9.292215146999167851860850719654

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