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Properties

Label 2-840-105.104-c1-0-0
Degree $2$
Conductor $840$
Sign $-0.341 - 0.939i$
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  + (−1.64 + 0.539i)3-s + (−1.30 − 1.81i)5-s + (−2.19 − 1.47i)7-s + (2.41 − 1.77i)9-s + 0.958i·11-s + 0.157·13-s + (3.12 + 2.29i)15-s − 2.51i·17-s + 1.98i·19-s + (4.41 + 1.23i)21-s − 2.67·23-s + (−1.60 + 4.73i)25-s + (−3.02 + 4.22i)27-s + 1.25i·29-s + 8.66i·31-s + ⋯
L(s)  = 1  + (−0.950 + 0.311i)3-s + (−0.582 − 0.812i)5-s + (−0.831 − 0.555i)7-s + (0.806 − 0.591i)9-s + 0.288i·11-s + 0.0436·13-s + (0.806 + 0.591i)15-s − 0.609i·17-s + 0.454i·19-s + (0.963 + 0.269i)21-s − 0.557·23-s + (−0.321 + 0.946i)25-s + (−0.582 + 0.813i)27-s + 0.232i·29-s + 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.341 - 0.939i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.341 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185945 + 0.265526i\)
\(L(\frac12)\) \(\approx\) \(0.185945 + 0.265526i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 0.539i)T \)
5 \( 1 + (1.30 + 1.81i)T \)
7 \( 1 + (2.19 + 1.47i)T \)
good11 \( 1 - 0.958iT - 11T^{2} \)
13 \( 1 - 0.157T + 13T^{2} \)
17 \( 1 + 2.51iT - 17T^{2} \)
19 \( 1 - 1.98iT - 19T^{2} \)
23 \( 1 + 2.67T + 23T^{2} \)
29 \( 1 - 1.25iT - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 - 2.29iT - 37T^{2} \)
41 \( 1 - 4.74T + 41T^{2} \)
43 \( 1 - 6.58iT - 43T^{2} \)
47 \( 1 + 5.60iT - 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 4.27iT - 61T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 + 4.43T + 73T^{2} \)
79 \( 1 - 0.517T + 79T^{2} \)
83 \( 1 - 18.1iT - 83T^{2} \)
89 \( 1 + 0.954T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−10.42965880686971597854318494496, −9.720009487984266202798268163571, −8.960308984153636824281024985675, −7.77716109420535470066162269849, −6.96476853593091384266300326005, −6.07610750964101946343148485239, −5.03942454671335980078937245879, −4.28716778768862804320201261759, −3.33339765203168010217672050605, −1.19652662058963379487213519963, 0.20127077126128006541075726348, 2.23366562150340092297221948852, 3.47868526499918746251576114957, 4.53542131103870693922083541535, 5.97164196720111378706901905181, 6.22219285419595182877553902082, 7.29097955451719295048116880925, 8.001758059149564321411298647188, 9.237272750139220353722953892632, 10.12864582685756942551923732070

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