Properties

Label 2-140-28.3-c1-0-2
Degree $2$
Conductor $140$
Sign $-0.155 - 0.987i$
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  + (−1.27 + 0.620i)2-s + (−0.331 + 0.573i)3-s + (1.23 − 1.57i)4-s + (−0.866 + 0.5i)5-s + (0.0650 − 0.934i)6-s + (1.68 + 2.03i)7-s + (−0.585 + 2.76i)8-s + (1.28 + 2.21i)9-s + (0.790 − 1.17i)10-s + (−3.12 − 1.80i)11-s + (0.496 + 1.22i)12-s + 5.83i·13-s + (−3.40 − 1.54i)14-s − 0.662i·15-s + (−0.971 − 3.88i)16-s + (−1.18 − 0.684i)17-s + ⋯
L(s)  = 1  + (−0.898 + 0.438i)2-s + (−0.191 + 0.331i)3-s + (0.615 − 0.788i)4-s + (−0.387 + 0.223i)5-s + (0.0265 − 0.381i)6-s + (0.637 + 0.770i)7-s + (−0.207 + 0.978i)8-s + (0.426 + 0.739i)9-s + (0.249 − 0.370i)10-s + (−0.943 − 0.544i)11-s + (0.143 + 0.354i)12-s + 1.61i·13-s + (−0.911 − 0.412i)14-s − 0.171i·15-s + (−0.242 − 0.970i)16-s + (−0.287 − 0.165i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429729 + 0.502457i\)
\(L(\frac12)\) \(\approx\) \(0.429729 + 0.502457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.620i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-1.68 - 2.03i)T \)
good3 \( 1 + (0.331 - 0.573i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.12 + 1.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.83iT - 13T^{2} \)
17 \( 1 + (1.18 + 0.684i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.04 - 3.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.81 + 1.62i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + (-4.43 + 7.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.39 + 9.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.832iT - 41T^{2} \)
43 \( 1 + 3.10iT - 43T^{2} \)
47 \( 1 + (-3.44 - 5.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.70 + 6.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.73 - 6.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 + 0.742i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.19 + 1.26i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.52iT - 71T^{2} \)
73 \( 1 + (4.47 + 2.58i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.82 + 5.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.49T + 83T^{2} \)
89 \( 1 + (-8.13 + 4.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.343iT - 97T^{2} \)
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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

−13.72129936115081069986972597263, −12.00318013990465754771492973335, −11.18911309388176204591734568704, −10.38981687988725798409625443039, −9.171110431016083641940245214174, −8.202900965750597166751472683145, −7.26146889021893056004942462102, −5.84398153348070646060525552190, −4.66882494005313933162087658763, −2.22467448367355201241763399500, 0.979798289880133223118971040087, 3.15412406017387629948178650013, 4.86586028865057141587915270550, 6.84376475237165648654058418335, 7.67626344895164899478811944619, 8.563895097832375659844534326054, 10.04096036903949313605887241067, 10.66572398350288176189847199196, 11.81553753186252686640181186241, 12.66258161134400617275120154437

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