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Properties

Label 2-840-35.13-c1-0-15
Degree $2$
Conductor $840$
Sign $0.0282 + 0.999i$
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  + (−0.707 + 0.707i)3-s + (−0.503 − 2.17i)5-s + (0.918 + 2.48i)7-s − 1.00i·9-s − 5.08·11-s + (3.79 − 3.79i)13-s + (1.89 + 1.18i)15-s + (−2.83 − 2.83i)17-s + 3.66·19-s + (−2.40 − 1.10i)21-s + (−0.591 − 0.591i)23-s + (−4.49 + 2.19i)25-s + (0.707 + 0.707i)27-s − 1.05i·29-s − 8.77i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.225 − 0.974i)5-s + (0.347 + 0.937i)7-s − 0.333i·9-s − 1.53·11-s + (1.05 − 1.05i)13-s + (0.489 + 0.305i)15-s + (−0.687 − 0.687i)17-s + 0.840·19-s + (−0.524 − 0.241i)21-s + (−0.123 − 0.123i)23-s + (−0.898 + 0.438i)25-s + (0.136 + 0.136i)27-s − 0.195i·29-s − 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0282 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0282 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0282 + 0.999i$
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: \((2,\ 840,\ (\ :1/2),\ 0.0282 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653950 - 0.635737i\)
\(L(\frac12)\) \(\approx\) \(0.653950 - 0.635737i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.503 + 2.17i)T \)
7 \( 1 + (-0.918 - 2.48i)T \)
good11 \( 1 + 5.08T + 11T^{2} \)
13 \( 1 + (-3.79 + 3.79i)T - 13iT^{2} \)
17 \( 1 + (2.83 + 2.83i)T + 17iT^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + (0.591 + 0.591i)T + 23iT^{2} \)
29 \( 1 + 1.05iT - 29T^{2} \)
31 \( 1 + 8.77iT - 31T^{2} \)
37 \( 1 + (-4.95 + 4.95i)T - 37iT^{2} \)
41 \( 1 - 2.91iT - 41T^{2} \)
43 \( 1 + (6.86 + 6.86i)T + 43iT^{2} \)
47 \( 1 + (1.97 + 1.97i)T + 47iT^{2} \)
53 \( 1 + (5.54 + 5.54i)T + 53iT^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 + 12.3iT - 61T^{2} \)
67 \( 1 + (-4.30 + 4.30i)T - 67iT^{2} \)
71 \( 1 - 9.87T + 71T^{2} \)
73 \( 1 + (-7.02 + 7.02i)T - 73iT^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 + (-5.33 + 5.33i)T - 83iT^{2} \)
89 \( 1 - 1.75T + 89T^{2} \)
97 \( 1 + (3.26 + 3.26i)T + 97iT^{2} \)
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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

−9.894833107919660862349384517294, −9.200231637557941125868280295836, −8.185899794798172534124700208213, −7.83945659641808363948051068881, −6.20444464551214749565744077354, −5.29582080694008838532728025106, −4.99270305900385259354918671818, −3.60663907027120470423071156682, −2.31004967719471751423629931678, −0.47930010095558400626313875700, 1.50977680215669077055570297446, 2.92578073951619184444863688476, 4.04832830593426799714111338902, 5.10557640906220082886356698583, 6.30443431991873759113312146575, 6.92787213028978514700852431164, 7.75779219908361320631955273563, 8.441172999087711019411702928752, 9.845125060273237767167229737313, 10.67890043800014466617617801865

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