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Properties

Label 2-840-280.237-c1-0-46
Degree $2$
Conductor $840$
Sign $0.0411 + 0.999i$
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  + (−1.38 + 0.288i)2-s + (−0.707 + 0.707i)3-s + (1.83 − 0.798i)4-s + (−1.97 − 1.04i)5-s + (0.775 − 1.18i)6-s + (0.379 + 2.61i)7-s + (−2.30 + 1.63i)8-s − 1.00i·9-s + (3.03 + 0.879i)10-s − 1.52i·11-s + (−0.731 + 1.86i)12-s + (−2.99 + 2.99i)13-s + (−1.28 − 3.51i)14-s + (2.13 − 0.656i)15-s + (2.72 − 2.92i)16-s + (0.848 − 0.848i)17-s + ⋯
L(s)  = 1  + (−0.978 + 0.203i)2-s + (−0.408 + 0.408i)3-s + (0.916 − 0.399i)4-s + (−0.883 − 0.468i)5-s + (0.316 − 0.482i)6-s + (0.143 + 0.989i)7-s + (−0.816 + 0.577i)8-s − 0.333i·9-s + (0.960 + 0.278i)10-s − 0.458i·11-s + (−0.211 + 0.537i)12-s + (−0.830 + 0.830i)13-s + (−0.342 − 0.939i)14-s + (0.551 − 0.169i)15-s + (0.681 − 0.732i)16-s + (0.205 − 0.205i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0411 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0411 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0411 + 0.999i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.0411 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210142 - 0.201661i\)
\(L(\frac12)\) \(\approx\) \(0.210142 - 0.201661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.288i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.97 + 1.04i)T \)
7 \( 1 + (-0.379 - 2.61i)T \)
good11 \( 1 + 1.52iT - 11T^{2} \)
13 \( 1 + (2.99 - 2.99i)T - 13iT^{2} \)
17 \( 1 + (-0.848 + 0.848i)T - 17iT^{2} \)
19 \( 1 - 1.11iT - 19T^{2} \)
23 \( 1 + (-2.90 + 2.90i)T - 23iT^{2} \)
29 \( 1 + 6.35T + 29T^{2} \)
31 \( 1 + 3.55iT - 31T^{2} \)
37 \( 1 + (-0.232 + 0.232i)T - 37iT^{2} \)
41 \( 1 + 6.76iT - 41T^{2} \)
43 \( 1 + (7.11 + 7.11i)T + 43iT^{2} \)
47 \( 1 + (1.87 - 1.87i)T - 47iT^{2} \)
53 \( 1 + (-0.0961 - 0.0961i)T + 53iT^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 - 9.42T + 61T^{2} \)
67 \( 1 + (-11.4 + 11.4i)T - 67iT^{2} \)
71 \( 1 + 4.00T + 71T^{2} \)
73 \( 1 + (10.1 + 10.1i)T + 73iT^{2} \)
79 \( 1 + 3.22iT - 79T^{2} \)
83 \( 1 + (2.36 - 2.36i)T - 83iT^{2} \)
89 \( 1 - 7.10T + 89T^{2} \)
97 \( 1 + (-5.73 + 5.73i)T - 97iT^{2} \)
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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

−9.774420123243189342722598796230, −9.096838498263981651181097740042, −8.484985795198978948798901777876, −7.57582464753826437327762029604, −6.68612364901544694813230548178, −5.59890672418144317609062413780, −4.84730708897424412165043283875, −3.44987667483528411797895473039, −2.03453499449466153268147202414, −0.23001951337318090527942613749, 1.18129056078849111586840192039, 2.76132402799826624487383713467, 3.82270601817920238917996425655, 5.12169745671303759318191475473, 6.53242429520540649303989745364, 7.32051122720111199600856652569, 7.63242628246394146464058512569, 8.540489739979856163101433206411, 9.871688626212639672920958237411, 10.31838556596677303955110209811

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