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Properties

Label 2-2340-39.2-c1-0-9
Degree $2$
Conductor $2340$
Sign $0.791 - 0.611i$
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 + (4.91 − 1.31i)7-s + (−3.41 − 0.913i)11-s + (3.41 − 1.16i)13-s + (−3.88 + 6.72i)17-s + (1.75 + 6.56i)19-s + (3.13 + 5.42i)23-s + 1.00i·25-s + (1.62 − 0.937i)29-s + (−1.00 + 1.00i)31-s + (4.40 + 2.54i)35-s + (−1.83 + 6.84i)37-s + (1.55 − 5.79i)41-s + (−6.96 − 4.02i)43-s + (6.57 − 6.57i)47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (1.85 − 0.497i)7-s + (−1.02 − 0.275i)11-s + (0.946 − 0.324i)13-s + (−0.942 + 1.63i)17-s + (0.403 + 1.50i)19-s + (0.653 + 1.13i)23-s + 0.200i·25-s + (0.301 − 0.174i)29-s + (−0.180 + 0.180i)31-s + (0.744 + 0.430i)35-s + (−0.301 + 1.12i)37-s + (0.242 − 0.904i)41-s + (−1.06 − 0.613i)43-s + (0.959 − 0.959i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.791 - 0.611i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.791 - 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.304077546\)
\(L(\frac12)\) \(\approx\) \(2.304077546\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 + 1.16i)T \)
good7 \( 1 + (-4.91 + 1.31i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.41 + 0.913i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.88 - 6.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.75 - 6.56i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.13 - 5.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.62 + 0.937i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.00 - 1.00i)T - 31iT^{2} \)
37 \( 1 + (1.83 - 6.84i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.55 + 5.79i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.96 + 4.02i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.57 + 6.57i)T - 47iT^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + (2.73 + 10.1i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.963 - 1.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-14.0 - 3.75i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.328 + 0.0880i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.98 + 1.98i)T + 73iT^{2} \)
79 \( 1 + 5.53T + 79T^{2} \)
83 \( 1 + (-9.81 - 9.81i)T + 83iT^{2} \)
89 \( 1 + (6.68 + 1.79i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.47 + 16.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−8.760501316838769539241401180738, −8.221791500801167649766242329084, −7.77583399044376512811982617922, −6.80940213876300413178046011656, −5.70712235024193875478082291919, −5.30278795910266196773290009990, −4.19420440815592901190698406575, −3.44067474032416590067965353417, −2.00360763406509366089786766926, −1.33531468887092804660378555588, 0.862325343301061345333045559284, 2.13598651314672016922202163729, 2.76829950072622560466394225804, 4.52490688010424516576831070588, 4.84928550965754246273283489802, 5.49396210491841775145871018401, 6.68044482525561402192868281450, 7.43561563406892433713338747998, 8.266535452135831011652113431578, 8.864818248220899563126608755971

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