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Properties

Label 2-840-840.53-c1-0-172
Degree $2$
Conductor $840$
Sign $-0.983 + 0.179i$
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  + (−1.40 + 0.126i)2-s + (1.11 − 1.32i)3-s + (1.96 − 0.355i)4-s + (1.61 − 1.54i)5-s + (−1.39 + 2.01i)6-s + (−2.64 − 0.163i)7-s + (−2.72 + 0.749i)8-s + (−0.534 − 2.95i)9-s + (−2.07 + 2.38i)10-s + (−2.26 + 3.92i)11-s + (1.71 − 3.01i)12-s + (0.229 − 0.229i)13-s + (3.74 − 0.103i)14-s + (−0.262 − 3.86i)15-s + (3.74 − 1.39i)16-s + (−6.05 + 1.62i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0892i)2-s + (0.641 − 0.767i)3-s + (0.984 − 0.177i)4-s + (0.722 − 0.691i)5-s + (−0.569 + 0.821i)6-s + (−0.998 − 0.0617i)7-s + (−0.964 + 0.264i)8-s + (−0.178 − 0.983i)9-s + (−0.657 + 0.753i)10-s + (−0.682 + 1.18i)11-s + (0.494 − 0.869i)12-s + (0.0636 − 0.0636i)13-s + (0.999 − 0.0275i)14-s + (−0.0678 − 0.997i)15-s + (0.936 − 0.349i)16-s + (−1.46 + 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.983 + 0.179i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0550298 - 0.606801i\)
\(L(\frac12)\) \(\approx\) \(0.0550298 - 0.606801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.126i)T \)
3 \( 1 + (-1.11 + 1.32i)T \)
5 \( 1 + (-1.61 + 1.54i)T \)
7 \( 1 + (2.64 + 0.163i)T \)
good11 \( 1 + (2.26 - 3.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.229 + 0.229i)T - 13iT^{2} \)
17 \( 1 + (6.05 - 1.62i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.28 + 5.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.55 + 1.48i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.429iT - 29T^{2} \)
31 \( 1 + (-2.50 + 4.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.269 - 1.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 + (1.23 - 1.23i)T - 43iT^{2} \)
47 \( 1 + (-1.96 + 7.33i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-11.2 + 3.00i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.56 + 0.902i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.1 + 6.99i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.44 + 0.386i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + (13.7 - 3.68i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.85 - 2.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.47 + 4.47i)T + 83iT^{2} \)
89 \( 1 + (4.03 + 6.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.18 + 5.18i)T - 97iT^{2} \)
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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

−9.829259371870028501094488221852, −8.767447474550440940834144641539, −8.443597299850765704695262398277, −7.17430443509486664637558514734, −6.65786361789969091278555605044, −5.81758915387087690466553279532, −4.28852433309365830327004410302, −2.52813616867647654584708993080, −2.06394331932831917646689257948, −0.33327187987843865047817979899, 2.16201192198520186196994391854, 2.93541486005199357559390968095, 3.85787565386650423355434058876, 5.68729469568017975447359071048, 6.29628068765250512828179543746, 7.33686148936663963443487369595, 8.403834958136664898640362339805, 8.965008001113818112171992438892, 9.804554676290298121506521997675, 10.46951071170386067059662618371

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