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Properties

Label 2-840-56.27-c1-0-25
Degree $2$
Conductor $840$
Sign $0.707 - 0.706i$
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  + (−0.718 + 1.21i)2-s − i·3-s + (−0.967 − 1.75i)4-s + 5-s + (1.21 + 0.718i)6-s + (1.92 + 1.81i)7-s + (2.82 + 0.0790i)8-s − 9-s + (−0.718 + 1.21i)10-s − 0.231·11-s + (−1.75 + 0.967i)12-s + 2.32·13-s + (−3.59 + 1.03i)14-s − i·15-s + (−2.12 + 3.38i)16-s + 1.23i·17-s + ⋯
L(s)  = 1  + (−0.508 + 0.861i)2-s − 0.577i·3-s + (−0.483 − 0.875i)4-s + 0.447·5-s + (0.497 + 0.293i)6-s + (0.726 + 0.687i)7-s + (0.999 + 0.0279i)8-s − 0.333·9-s + (−0.227 + 0.385i)10-s − 0.0699·11-s + (−0.505 + 0.279i)12-s + 0.643·13-s + (−0.960 + 0.276i)14-s − 0.258i·15-s + (−0.531 + 0.846i)16-s + 0.299i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.707 - 0.706i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.707 - 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26205 + 0.522637i\)
\(L(\frac12)\) \(\approx\) \(1.26205 + 0.522637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.718 - 1.21i)T \)
3 \( 1 + iT \)
5 \( 1 - T \)
7 \( 1 + (-1.92 - 1.81i)T \)
good11 \( 1 + 0.231T + 11T^{2} \)
13 \( 1 - 2.32T + 13T^{2} \)
17 \( 1 - 1.23iT - 17T^{2} \)
19 \( 1 - 2.55iT - 19T^{2} \)
23 \( 1 - 0.838iT - 23T^{2} \)
29 \( 1 + 7.90iT - 29T^{2} \)
31 \( 1 - 4.01T + 31T^{2} \)
37 \( 1 - 2.49iT - 37T^{2} \)
41 \( 1 - 6.89iT - 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 0.334T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 - 10.0iT - 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 - 3.45iT - 71T^{2} \)
73 \( 1 + 7.22iT - 73T^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 - 4.60iT - 83T^{2} \)
89 \( 1 - 0.827iT - 89T^{2} \)
97 \( 1 + 9.30iT - 97T^{2} \)
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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

−10.08138638580497428991380070489, −9.296738897665030889785682233606, −8.275675617471160200019868666703, −8.039286750454511566301057429700, −6.82425545774781311730550036082, −5.99987076835223251955098188830, −5.42998921036454126414045184007, −4.22352775510135031263876314423, −2.35987196882378603281109414336, −1.21803977841163865701185265930, 1.01274515438177804322235391847, 2.39073902611197845627450234998, 3.59838747431799254775389973523, 4.50013410570343320154304068839, 5.38346965075512187407561582240, 6.84473279250512579883121770329, 7.77718013808022418290180921623, 8.727270280390871312910369188425, 9.278389397931744912360399563876, 10.31090567450139935472671049013

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