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Properties

Label 2-840-168.101-c1-0-127
Degree $2$
Conductor $840$
Sign $0.370 - 0.928i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (0.331 − 1.37i)2-s + (−0.303 − 1.70i)3-s + (−1.78 − 0.910i)4-s + (0.866 − 0.5i)5-s + (−2.44 − 0.147i)6-s + (−2.64 + 0.0708i)7-s + (−1.84 + 2.14i)8-s + (−2.81 + 1.03i)9-s + (−0.400 − 1.35i)10-s + (−0.735 + 1.27i)11-s + (−1.01 + 3.31i)12-s + 1.58·13-s + (−0.778 + 3.65i)14-s + (−1.11 − 1.32i)15-s + (2.34 + 3.24i)16-s + (−2.77 + 4.81i)17-s + ⋯
L(s)  = 1  + (0.234 − 0.972i)2-s + (−0.175 − 0.984i)3-s + (−0.890 − 0.455i)4-s + (0.387 − 0.223i)5-s + (−0.998 − 0.0603i)6-s + (−0.999 + 0.0267i)7-s + (−0.651 + 0.759i)8-s + (−0.938 + 0.344i)9-s + (−0.126 − 0.428i)10-s + (−0.221 + 0.384i)11-s + (−0.292 + 0.956i)12-s + 0.439·13-s + (−0.207 + 0.978i)14-s + (−0.287 − 0.342i)15-s + (0.585 + 0.810i)16-s + (−0.673 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.370 - 0.928i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116503 + 0.0789638i\)
\(L(\frac12)\) \(\approx\) \(0.116503 + 0.0789638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.331 + 1.37i)T \)
3 \( 1 + (0.303 + 1.70i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (2.64 - 0.0708i)T \)
good11 \( 1 + (0.735 - 1.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.58T + 13T^{2} \)
17 \( 1 + (2.77 - 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.80 + 4.85i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.32 - 0.762i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + (-0.436 - 0.252i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.430 - 0.248i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.40T + 41T^{2} \)
43 \( 1 - 0.257iT - 43T^{2} \)
47 \( 1 + (5.41 + 9.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.79 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.84 + 2.21i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.65 + 13.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.84 + 3.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.60iT - 71T^{2} \)
73 \( 1 + (-8.97 - 5.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.01 + 5.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−9.501967364656857775523572057721, −8.894185590104208817960506397840, −7.959093154390688334821768765382, −6.62132330902255215116125636389, −6.07838332601593868928187019257, −5.02337004265396092780418745775, −3.78316950781526712589453383666, −2.59963929692459260285666354907, −1.67194259004588504602185204167, −0.06240516116493112739048694490, 2.90940319807786763369986236779, 3.78977439627272319712850422359, 4.75030429747174863715848525798, 5.89008850609605218477748698265, 6.21614677404665407507242256306, 7.33868409469767941804684426729, 8.469552939066507349977943124909, 9.230269643458572376335228547351, 9.821020967750115777173154689184, 10.64170008486294822110793597879

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