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Properties

Label 2-840-35.27-c1-0-18
Degree $2$
Conductor $840$
Sign $0.634 + 0.772i$
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  + (−0.707 − 0.707i)3-s + (1.43 + 1.71i)5-s + (1.41 − 2.23i)7-s + 1.00i·9-s + 0.566·11-s + (−5.03 − 5.03i)13-s + (0.194 − 2.22i)15-s + (0.984 − 0.984i)17-s + 7.61·19-s + (−2.58 + 0.581i)21-s + (2.55 − 2.55i)23-s + (−0.865 + 4.92i)25-s + (0.707 − 0.707i)27-s + 7.85i·29-s − 7.09i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.642 + 0.765i)5-s + (0.534 − 0.845i)7-s + 0.333i·9-s + 0.170·11-s + (−1.39 − 1.39i)13-s + (0.0501 − 0.575i)15-s + (0.238 − 0.238i)17-s + 1.74·19-s + (−0.563 + 0.126i)21-s + (0.531 − 0.531i)23-s + (−0.173 + 0.984i)25-s + (0.136 − 0.136i)27-s + 1.45i·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39999 - 0.661522i\)
\(L(\frac12)\) \(\approx\) \(1.39999 - 0.661522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.43 - 1.71i)T \)
7 \( 1 + (-1.41 + 2.23i)T \)
good11 \( 1 - 0.566T + 11T^{2} \)
13 \( 1 + (5.03 + 5.03i)T + 13iT^{2} \)
17 \( 1 + (-0.984 + 0.984i)T - 17iT^{2} \)
19 \( 1 - 7.61T + 19T^{2} \)
23 \( 1 + (-2.55 + 2.55i)T - 23iT^{2} \)
29 \( 1 - 7.85iT - 29T^{2} \)
31 \( 1 + 7.09iT - 31T^{2} \)
37 \( 1 + (-0.887 - 0.887i)T + 37iT^{2} \)
41 \( 1 + 6.29iT - 41T^{2} \)
43 \( 1 + (2.74 - 2.74i)T - 43iT^{2} \)
47 \( 1 + (-3.25 + 3.25i)T - 47iT^{2} \)
53 \( 1 + (-7.27 + 7.27i)T - 53iT^{2} \)
59 \( 1 - 6.62T + 59T^{2} \)
61 \( 1 + 5.46iT - 61T^{2} \)
67 \( 1 + (2.53 + 2.53i)T + 67iT^{2} \)
71 \( 1 - 8.01T + 71T^{2} \)
73 \( 1 + (-8.97 - 8.97i)T + 73iT^{2} \)
79 \( 1 - 1.85iT - 79T^{2} \)
83 \( 1 + (2.61 + 2.61i)T + 83iT^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (4.32 - 4.32i)T - 97iT^{2} \)
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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

−10.13200486964713820227281641487, −9.567380318365657011647627889163, −8.107496447351391900548592283215, −7.29057953729453468729634403762, −6.91601934474526132769699230613, −5.49743917163894195769443126037, −5.09015870784968413369322538433, −3.47062829740713275985438877052, −2.41318056481942608146380207539, −0.898523226037773304543838168848, 1.41928708908368558719713478329, 2.64493889747689492745977808594, 4.27941672297678842853340823779, 5.10846917198347653891680155541, 5.63716523948563621706150515776, 6.78738024839593977913749838793, 7.82013710668559299163056573200, 8.947826818044774881630649419481, 9.464517363238404236655997411145, 10.04013436469216038236882966043

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