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Properties

Label 2-840-105.104-c1-0-1
Degree $2$
Conductor $840$
Sign $-0.0116 - 0.999i$
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  + (−0.773 − 1.54i)3-s + (0.194 − 2.22i)5-s + (−0.942 + 2.47i)7-s + (−1.80 + 2.39i)9-s + 1.45i·11-s − 4.42·13-s + (−3.60 + 1.42i)15-s + 0.440i·17-s + 6.54i·19-s + (4.56 − 0.450i)21-s − 2.43·23-s + (−4.92 − 0.864i)25-s + (5.10 + 0.944i)27-s + 4.30i·29-s − 2.86i·31-s + ⋯
L(s)  = 1  + (−0.446 − 0.894i)3-s + (0.0867 − 0.996i)5-s + (−0.356 + 0.934i)7-s + (−0.601 + 0.798i)9-s + 0.438i·11-s − 1.22·13-s + (−0.930 + 0.366i)15-s + 0.106i·17-s + 1.50i·19-s + (0.995 − 0.0984i)21-s − 0.508·23-s + (−0.984 − 0.172i)25-s + (0.983 + 0.181i)27-s + 0.800i·29-s − 0.514i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0116 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0116 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0116 - 0.999i$
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: \((2,\ 840,\ (\ :1/2),\ -0.0116 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307493 + 0.311102i\)
\(L(\frac12)\) \(\approx\) \(0.307493 + 0.311102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.773 + 1.54i)T \)
5 \( 1 + (-0.194 + 2.22i)T \)
7 \( 1 + (0.942 - 2.47i)T \)
good11 \( 1 - 1.45iT - 11T^{2} \)
13 \( 1 + 4.42T + 13T^{2} \)
17 \( 1 - 0.440iT - 17T^{2} \)
19 \( 1 - 6.54iT - 19T^{2} \)
23 \( 1 + 2.43T + 23T^{2} \)
29 \( 1 - 4.30iT - 29T^{2} \)
31 \( 1 + 2.86iT - 31T^{2} \)
37 \( 1 + 9.10iT - 37T^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 - 8.57iT - 43T^{2} \)
47 \( 1 - 6.31iT - 47T^{2} \)
53 \( 1 - 9.37T + 53T^{2} \)
59 \( 1 - 8.33T + 59T^{2} \)
61 \( 1 - 7.14iT - 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + 1.12iT - 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 6.83T + 79T^{2} \)
83 \( 1 + 5.75iT - 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 6.27T + 97T^{2} \)
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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

−10.28045821925788591054802358468, −9.549643089133426206333222593440, −8.619559227901826205870285845579, −7.87497150476944434084468533411, −7.01900475616318215168353362926, −5.81512692563289904286695120860, −5.44511331859163383845641864236, −4.27203793185500103607964625617, −2.56823751749929230156557697264, −1.57854158972146273182772163486, 0.21770932803427180746740216526, 2.63208728910444789140752877189, 3.55838302556247208109122971303, 4.54084022484246190027013012516, 5.50677193556141493101369760183, 6.69001639692002328446847082040, 7.08530037081039081781036463639, 8.352666455686452454498341438332, 9.530094793934449966291496916665, 10.08125475788962327373498822301

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