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Properties

Label 2-840-840.629-c1-0-174
Degree $2$
Conductor $840$
Sign $-0.944 + 0.329i$
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  + (1.29 + 0.560i)2-s + (−0.135 − 1.72i)3-s + (1.37 + 1.45i)4-s + (−1.93 + 1.11i)5-s + (0.791 − 2.31i)6-s + (−2.16 − 1.52i)7-s + (0.965 + 2.65i)8-s + (−2.96 + 0.469i)9-s + (−3.14 + 0.367i)10-s − 4.91·11-s + (2.32 − 2.56i)12-s − 3.98i·13-s + (−1.94 − 3.19i)14-s + (2.19 + 3.19i)15-s + (−0.236 + 3.99i)16-s − 0.676i·17-s + ⋯
L(s)  = 1  + (0.918 + 0.396i)2-s + (−0.0785 − 0.996i)3-s + (0.685 + 0.727i)4-s + (−0.865 + 0.500i)5-s + (0.322 − 0.946i)6-s + (−0.816 − 0.577i)7-s + (0.341 + 0.939i)8-s + (−0.987 + 0.156i)9-s + (−0.993 + 0.116i)10-s − 1.48·11-s + (0.671 − 0.740i)12-s − 1.10i·13-s + (−0.520 − 0.853i)14-s + (0.566 + 0.823i)15-s + (−0.0591 + 0.998i)16-s − 0.164i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0620971 - 0.366496i\)
\(L(\frac12)\) \(\approx\) \(0.0620971 - 0.366496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.560i)T \)
3 \( 1 + (0.135 + 1.72i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (2.16 + 1.52i)T \)
good11 \( 1 + 4.91T + 11T^{2} \)
13 \( 1 + 3.98iT - 13T^{2} \)
17 \( 1 + 0.676iT - 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
23 \( 1 + 1.13T + 23T^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 + 4.17iT - 31T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + 1.89T + 41T^{2} \)
43 \( 1 + 9.82T + 43T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 + 13.0iT - 53T^{2} \)
59 \( 1 - 5.42iT - 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 4.83T + 67T^{2} \)
71 \( 1 - 1.84iT - 71T^{2} \)
73 \( 1 + 9.95T + 73T^{2} \)
79 \( 1 + 9.58T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 5.53T + 89T^{2} \)
97 \( 1 + 7.68T + 97T^{2} \)
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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

−10.19010407470540927409366407235, −8.281993192854945593222295043259, −7.955613198023033961656779909218, −7.14303916468735347881752797017, −6.42705406126900556140434597032, −5.58487616859505739813013212945, −4.38524260176211973585191162811, −3.17595838251328714196812636754, −2.54902763135296061444673604111, −0.12155398592028614062307839645, 2.38838333721446153493802574302, 3.38879831056624896099693825164, 4.31535413531471080185704215356, 4.99206757483293474060505143537, 5.92175857522746909146155647989, 6.87182555311108930000834703819, 8.229430352704942176841207573823, 8.997988836592916757407687594339, 10.00908091283665415210304274258, 10.60937442988107332891622586887

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