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Properties

Label 2-840-280.107-c1-0-22
Degree $2$
Conductor $840$
Sign $0.123 - 0.992i$
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  + (1.25 + 0.650i)2-s + (−0.965 − 0.258i)3-s + (1.15 + 1.63i)4-s + (−2.12 − 0.709i)5-s + (−1.04 − 0.953i)6-s + (−0.607 − 2.57i)7-s + (0.384 + 2.80i)8-s + (0.866 + 0.499i)9-s + (−2.20 − 2.27i)10-s + (2.35 + 4.07i)11-s + (−0.690 − 1.87i)12-s + (1.44 + 1.44i)13-s + (0.912 − 3.62i)14-s + (1.86 + 1.23i)15-s + (−1.34 + 3.76i)16-s + (−0.672 + 2.50i)17-s + ⋯
L(s)  = 1  + (0.887 + 0.460i)2-s + (−0.557 − 0.149i)3-s + (0.576 + 0.817i)4-s + (−0.948 − 0.317i)5-s + (−0.426 − 0.389i)6-s + (−0.229 − 0.973i)7-s + (0.135 + 0.990i)8-s + (0.288 + 0.166i)9-s + (−0.695 − 0.718i)10-s + (0.709 + 1.22i)11-s + (−0.199 − 0.541i)12-s + (0.402 + 0.402i)13-s + (0.244 − 0.969i)14-s + (0.481 + 0.318i)15-s + (−0.335 + 0.942i)16-s + (−0.163 + 0.608i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.123 - 0.992i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.123 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34903 + 1.19213i\)
\(L(\frac12)\) \(\approx\) \(1.34903 + 1.19213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.650i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (2.12 + 0.709i)T \)
7 \( 1 + (0.607 + 2.57i)T \)
good11 \( 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.44 - 1.44i)T + 13iT^{2} \)
17 \( 1 + (0.672 - 2.50i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.85 - 3.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 - 1.39i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 + (-2.13 + 1.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.91 - 7.14i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.96T + 41T^{2} \)
43 \( 1 + (-1.62 + 1.62i)T - 43iT^{2} \)
47 \( 1 + (1.30 + 4.88i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.34 - 8.75i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-11.4 + 6.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.44 + 4.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.97 + 11.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.5iT - 71T^{2} \)
73 \( 1 + (12.8 + 3.44i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.51 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.03 - 3.03i)T - 83iT^{2} \)
89 \( 1 + (-0.0534 - 0.0308i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 - 8.57i)T + 97iT^{2} \)
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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

−10.55619150508723093492371663380, −9.680658853900044405919057639783, −8.292199219100251548550205804434, −7.59141353705738569443558296435, −6.88577842417745689472378286516, −6.15073774946364210539240232387, −4.85233014856434570852034445653, −4.19883877264914493851800490170, −3.47793541809706221501032242415, −1.49835273241165514946836161148, 0.77916152739501605291933767508, 2.76461784879666984810611086359, 3.49927037496417321879213959336, 4.56015587565258020075446823433, 5.58070374322163915845548205663, 6.25333702594590397386919894840, 7.13236483834102667442665498451, 8.364957572890127863662915121450, 9.303598774953448498290950371737, 10.30248868901398084906964934360

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