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Properties

Label 2-840-168.101-c1-0-74
Degree $2$
Conductor $840$
Sign $-0.640 + 0.767i$
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  + (−1.38 + 0.281i)2-s + (−1.72 − 0.101i)3-s + (1.84 − 0.780i)4-s + (−0.866 + 0.5i)5-s + (2.42 − 0.345i)6-s + (−0.542 − 2.58i)7-s + (−2.33 + 1.59i)8-s + (2.97 + 0.351i)9-s + (1.05 − 0.936i)10-s + (−0.823 + 1.42i)11-s + (−3.26 + 1.16i)12-s + 1.37·13-s + (1.48 + 3.43i)14-s + (1.54 − 0.776i)15-s + (2.78 − 2.87i)16-s + (1.71 − 2.97i)17-s + ⋯
L(s)  = 1  + (−0.980 + 0.198i)2-s + (−0.998 − 0.0586i)3-s + (0.920 − 0.390i)4-s + (−0.387 + 0.223i)5-s + (0.989 − 0.141i)6-s + (−0.205 − 0.978i)7-s + (−0.824 + 0.565i)8-s + (0.993 + 0.117i)9-s + (0.335 − 0.296i)10-s + (−0.248 + 0.430i)11-s + (−0.942 + 0.335i)12-s + 0.382·13-s + (0.395 + 0.918i)14-s + (0.399 − 0.200i)15-s + (0.695 − 0.718i)16-s + (0.416 − 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.640 + 0.767i$
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: \((2,\ 840,\ (\ :1/2),\ -0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115406 - 0.246520i\)
\(L(\frac12)\) \(\approx\) \(0.115406 - 0.246520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.281i)T \)
3 \( 1 + (1.72 + 0.101i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.542 + 2.58i)T \)
good11 \( 1 + (0.823 - 1.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 + (-1.71 + 2.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.30 - 2.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.30 - 1.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.197T + 29T^{2} \)
31 \( 1 + (2.59 + 1.49i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.72 - 1.57i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 - 2.90iT - 43T^{2} \)
47 \( 1 + (4.47 + 7.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.68 + 2.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.9 + 6.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.06 + 5.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.09 + 3.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.209iT - 71T^{2} \)
73 \( 1 + (14.6 + 8.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.67 + 4.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 + (0.829 + 1.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.837iT - 97T^{2} \)
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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

−10.05511915450724922934171651856, −9.272884737641891424080286423272, −7.82718103819121404368031090896, −7.46496698052981788807158547893, −6.60314673598154913135525429821, −5.77306700886066301150852813072, −4.62570791092007588131150314411, −3.35598870841989792941187929558, −1.57587711678542757890273538467, −0.22948650363124197122961812662, 1.33261605771369465400583693455, 2.84888790681078125430846388210, 4.13619528491911843956135539977, 5.58121578814436229108007017773, 6.10869179178488274923344550867, 7.16734023526063534218273420416, 8.071450017395724176888754874354, 8.910815695713212081179635926954, 9.644582666879179793538621613415, 10.65890853450299056640214441754

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