Properties

Label 2-840-105.104-c1-0-17
Degree $2$
Conductor $840$
Sign $0.951 - 0.306i$
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  + (−0.726 + 1.57i)3-s + (−2.20 − 0.341i)5-s + (−2.06 − 1.64i)7-s + (−1.94 − 2.28i)9-s − 1.06i·11-s + 4.82·13-s + (2.14 − 3.22i)15-s + 7.89i·17-s − 4.02i·19-s + (4.09 − 2.05i)21-s + 5.69·23-s + (4.76 + 1.50i)25-s + (5.00 − 1.40i)27-s + 2.00i·29-s − 4.89i·31-s + ⋯
L(s)  = 1  + (−0.419 + 0.907i)3-s + (−0.988 − 0.152i)5-s + (−0.781 − 0.623i)7-s + (−0.648 − 0.761i)9-s − 0.320i·11-s + 1.33·13-s + (0.552 − 0.833i)15-s + 1.91i·17-s − 0.922i·19-s + (0.893 − 0.448i)21-s + 1.18·23-s + (0.953 + 0.301i)25-s + (0.962 − 0.269i)27-s + 0.372i·29-s − 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.951 - 0.306i$
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: \((2,\ 840,\ (\ :1/2),\ 0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957758 + 0.150528i\)
\(L(\frac12)\) \(\approx\) \(0.957758 + 0.150528i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.726 - 1.57i)T \)
5 \( 1 + (2.20 + 0.341i)T \)
7 \( 1 + (2.06 + 1.64i)T \)
good11 \( 1 + 1.06iT - 11T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 - 7.89iT - 17T^{2} \)
19 \( 1 + 4.02iT - 19T^{2} \)
23 \( 1 - 5.69T + 23T^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + 2.56iT - 37T^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 - 6.15iT - 43T^{2} \)
47 \( 1 - 2.27iT - 47T^{2} \)
53 \( 1 - 9.84T + 53T^{2} \)
59 \( 1 + 5.87T + 59T^{2} \)
61 \( 1 + 7.02iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 0.0512iT - 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 - 7.00T + 79T^{2} \)
83 \( 1 + 7.59iT - 83T^{2} \)
89 \( 1 - 9.72T + 89T^{2} \)
97 \( 1 + 1.77T + 97T^{2} \)
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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

−10.62195505501770349005937069721, −9.303153354045109551602197835138, −8.735684551253562846387036842804, −7.78863221288263393999106538518, −6.58948173773911539629128432122, −5.95521519001421043759284711035, −4.64743779800719060651801197273, −3.82530933738351017127731350330, −3.26126883567804106209310218561, −0.77519926117922354515388047905, 0.862509350061340601957447676348, 2.60889436895321989979620317210, 3.55493191309172997624333697747, 4.96239706872142362646694062452, 5.90965394256081618818318492484, 6.86169652119606730291407490739, 7.37944503983509008168096250592, 8.454232352197533363611070245639, 9.075951381025776841970265223563, 10.32832378265692564563790490742

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