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Properties

Label 2-840-35.13-c1-0-5
Degree $2$
Conductor $840$
Sign $-0.208 - 0.977i$
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 + (−0.793 + 2.09i)5-s + (2.38 + 1.13i)7-s − 1.00i·9-s + 1.83·11-s + (1.48 − 1.48i)13-s + (−0.917 − 2.03i)15-s + (4.65 + 4.65i)17-s − 3.78·19-s + (−2.49 + 0.885i)21-s + (0.980 + 0.980i)23-s + (−3.74 − 3.31i)25-s + (0.707 + 0.707i)27-s + 3.98i·29-s + 0.418i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.354 + 0.934i)5-s + (0.902 + 0.429i)7-s − 0.333i·9-s + 0.552·11-s + (0.412 − 0.412i)13-s + (−0.236 − 0.526i)15-s + (1.12 + 1.12i)17-s − 0.868·19-s + (−0.544 + 0.193i)21-s + (0.204 + 0.204i)23-s + (−0.748 − 0.663i)25-s + (0.136 + 0.136i)27-s + 0.740i·29-s + 0.0752i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.208 - 0.977i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.208 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847028 + 1.04715i\)
\(L(\frac12)\) \(\approx\) \(0.847028 + 1.04715i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.793 - 2.09i)T \)
7 \( 1 + (-2.38 - 1.13i)T \)
good11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + (-1.48 + 1.48i)T - 13iT^{2} \)
17 \( 1 + (-4.65 - 4.65i)T + 17iT^{2} \)
19 \( 1 + 3.78T + 19T^{2} \)
23 \( 1 + (-0.980 - 0.980i)T + 23iT^{2} \)
29 \( 1 - 3.98iT - 29T^{2} \)
31 \( 1 - 0.418iT - 31T^{2} \)
37 \( 1 + (3.70 - 3.70i)T - 37iT^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + (6.57 + 6.57i)T + 43iT^{2} \)
47 \( 1 + (5.32 + 5.32i)T + 47iT^{2} \)
53 \( 1 + (-3.94 - 3.94i)T + 53iT^{2} \)
59 \( 1 + 7.45T + 59T^{2} \)
61 \( 1 - 9.97iT - 61T^{2} \)
67 \( 1 + (-5.40 + 5.40i)T - 67iT^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + (-5.08 + 5.08i)T - 73iT^{2} \)
79 \( 1 - 6.33iT - 79T^{2} \)
83 \( 1 + (5.50 - 5.50i)T - 83iT^{2} \)
89 \( 1 + 6.27T + 89T^{2} \)
97 \( 1 + (-2.50 - 2.50i)T + 97iT^{2} \)
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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

−10.58141082241416604730066512389, −9.831413440970381210650267893113, −8.567709612317100759038670367364, −8.049877783716741761788346675542, −6.88915616973668576528025365560, −6.07869002671880454687308926841, −5.17854361157875835249514873471, −4.03310649130606049038297620124, −3.17185812407836332191387231652, −1.60405075349851877930639908876, 0.75934827117409413179682517513, 1.87246973057683678979650876805, 3.72276408381544431717323923330, 4.67500995456468528611132917321, 5.39645788734980787944862146525, 6.54153342125752910405697815471, 7.50089402351074665104343342515, 8.192307907789448744865730406127, 8.998579700784122933719399002905, 9.941556933849841029996981739865

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