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Properties

Label 2-840-105.104-c1-0-15
Degree $2$
Conductor $840$
Sign $0.685 - 0.728i$
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  + (1.70 − 0.301i)3-s + (−2.04 − 0.902i)5-s + (−1.30 + 2.30i)7-s + (2.81 − 1.02i)9-s + 5.71i·11-s + 3.76·13-s + (−3.76 − 0.922i)15-s + 3.22i·17-s + 0.786i·19-s + (−1.52 + 4.32i)21-s + 1.52·23-s + (3.37 + 3.69i)25-s + (4.49 − 2.60i)27-s − 6.77i·29-s − 1.56i·31-s + ⋯
L(s)  = 1  + (0.984 − 0.174i)3-s + (−0.914 − 0.403i)5-s + (−0.492 + 0.870i)7-s + (0.939 − 0.343i)9-s + 1.72i·11-s + 1.04·13-s + (−0.971 − 0.238i)15-s + 0.783i·17-s + 0.180i·19-s + (−0.333 + 0.942i)21-s + 0.318·23-s + (0.674 + 0.738i)25-s + (0.865 − 0.501i)27-s − 1.25i·29-s − 0.281i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.685 - 0.728i$
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: \((2,\ 840,\ (\ :1/2),\ 0.685 - 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64099 + 0.709063i\)
\(L(\frac12)\) \(\approx\) \(1.64099 + 0.709063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.301i)T \)
5 \( 1 + (2.04 + 0.902i)T \)
7 \( 1 + (1.30 - 2.30i)T \)
good11 \( 1 - 5.71iT - 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 - 3.22iT - 17T^{2} \)
19 \( 1 - 0.786iT - 19T^{2} \)
23 \( 1 - 1.52T + 23T^{2} \)
29 \( 1 + 6.77iT - 29T^{2} \)
31 \( 1 + 1.56iT - 31T^{2} \)
37 \( 1 - 8.83iT - 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 - 8.85iT - 43T^{2} \)
47 \( 1 - 1.75iT - 47T^{2} \)
53 \( 1 - 0.616T + 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + 6.26iT - 67T^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 10.5iT - 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 - 7.69T + 97T^{2} \)
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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

−9.960080136373969093506936333769, −9.437010125201940634946787923022, −8.473104237310004353634187901452, −7.996698648093624194986904824536, −7.03280986694796293174608877212, −6.10884349694298131375899549059, −4.64619331079073507695701658759, −3.91850299732674291607434993478, −2.81589669317432353458686954693, −1.58289126492988236099634042266, 0.857080206798748984401771688319, 2.93253448571495920401786882179, 3.52168932778394058603060687013, 4.26275149698278660142847298156, 5.76320898063921298720067939992, 7.00862475722078021441221981483, 7.45089586032138453636327402746, 8.690329038762788676150661063659, 8.820057648290523594647839769958, 10.23076772998305410472686052134

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