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Properties

Label 2-140-140.67-c1-0-12
Degree $2$
Conductor $140$
Sign $0.485 + 0.874i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (1.20 − 0.739i)2-s + (−2.47 + 0.664i)3-s + (0.906 − 1.78i)4-s + (1.93 − 1.12i)5-s + (−2.49 + 2.63i)6-s + (1.41 − 2.23i)7-s + (−0.224 − 2.81i)8-s + (3.10 − 1.79i)9-s + (1.50 − 2.78i)10-s + (1.59 + 0.921i)11-s + (−1.06 + 5.02i)12-s + (−2.94 + 2.94i)13-s + (0.0455 − 3.74i)14-s + (−4.04 + 4.06i)15-s + (−2.35 − 3.23i)16-s + (−2.96 + 0.795i)17-s + ⋯
L(s)  = 1  + (0.852 − 0.522i)2-s + (−1.43 + 0.383i)3-s + (0.453 − 0.891i)4-s + (0.864 − 0.502i)5-s + (−1.01 + 1.07i)6-s + (0.533 − 0.846i)7-s + (−0.0793 − 0.996i)8-s + (1.03 − 0.597i)9-s + (0.474 − 0.880i)10-s + (0.481 + 0.277i)11-s + (−0.307 + 1.44i)12-s + (−0.817 + 0.817i)13-s + (0.0121 − 0.999i)14-s + (−1.04 + 1.05i)15-s + (−0.588 − 0.808i)16-s + (−0.720 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13042 - 0.665172i\)
\(L(\frac12)\) \(\approx\) \(1.13042 - 0.665172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.739i)T \)
5 \( 1 + (-1.93 + 1.12i)T \)
7 \( 1 + (-1.41 + 2.23i)T \)
good3 \( 1 + (2.47 - 0.664i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.59 - 0.921i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 - 2.94i)T - 13iT^{2} \)
17 \( 1 + (2.96 - 0.795i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.66 - 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.654 - 2.44i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.35iT - 29T^{2} \)
31 \( 1 + (3.78 + 2.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.03 - 3.86i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (1.02 + 1.02i)T + 43iT^{2} \)
47 \( 1 + (0.785 + 0.210i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.699 - 2.61i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.11 - 3.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.856 - 3.19i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (0.529 + 1.97i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.95 + 6.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.227 + 0.227i)T + 83iT^{2} \)
89 \( 1 + (-3.75 + 2.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.196 - 0.196i)T + 97iT^{2} \)
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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

−12.76347932826955610479946233491, −11.94912979728148724479542660361, −11.10709482440967568598076430887, −10.24444054908187752392890024494, −9.412612075495199496557969838538, −7.09018559268813180088952958114, −6.00735191197833144047256702757, −5.01963416766893849673308448948, −4.21314673332037128913578067026, −1.55662292501773840396882660450, 2.51586020272795532032845045960, 4.86319539583528175839391448682, 5.67458714717837303714490299410, 6.44373602560960798029188214629, 7.49593682139453478195936138593, 9.145614226640952286169073350308, 10.78892440245141706685324399422, 11.49482234328125693663165065010, 12.35569036260314003511951772440, 13.20730744233662034031089983772

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