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Properties

Label 2-140-28.19-c1-0-3
Degree $2$
Conductor $140$
Sign $-0.629 - 0.777i$
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  + (0.0915 + 1.41i)2-s + (1.49 + 2.59i)3-s + (−1.98 + 0.258i)4-s + (−0.866 − 0.5i)5-s + (−3.52 + 2.35i)6-s + (2.06 − 1.65i)7-s + (−0.546 − 2.77i)8-s + (−2.99 + 5.18i)9-s + (0.626 − 1.26i)10-s + (1.93 − 1.11i)11-s + (−3.64 − 4.76i)12-s − 3.17i·13-s + (2.52 + 2.75i)14-s − 2.99i·15-s + (3.86 − 1.02i)16-s + (−2.98 + 1.72i)17-s + ⋯
L(s)  = 1  + (0.0647 + 0.997i)2-s + (0.865 + 1.49i)3-s + (−0.991 + 0.129i)4-s + (−0.387 − 0.223i)5-s + (−1.43 + 0.960i)6-s + (0.778 − 0.627i)7-s + (−0.193 − 0.981i)8-s + (−0.998 + 1.72i)9-s + (0.198 − 0.400i)10-s + (0.584 − 0.337i)11-s + (−1.05 − 1.37i)12-s − 0.879i·13-s + (0.676 + 0.736i)14-s − 0.774i·15-s + (0.966 − 0.256i)16-s + (−0.723 + 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.629 - 0.777i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.629 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560978 + 1.17598i\)
\(L(\frac12)\) \(\approx\) \(0.560978 + 1.17598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0915 - 1.41i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.06 + 1.65i)T \)
good3 \( 1 + (-1.49 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.93 + 1.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.17iT - 13T^{2} \)
17 \( 1 + (2.98 - 1.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.02 - 1.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.30 - 1.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.38T + 29T^{2} \)
31 \( 1 + (-2.44 - 4.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.59 + 9.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 + 9.95iT - 43T^{2} \)
47 \( 1 + (3.06 - 5.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.32 + 4.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 + 1.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0456 + 0.0263i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.212iT - 71T^{2} \)
73 \( 1 + (12.8 - 7.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.399 + 0.230i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (-6.07 - 3.51i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.185iT - 97T^{2} \)
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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

−14.00354466009090905813929813779, −12.92971597215409315762441681752, −11.15228878360838373288713932403, −10.23507606179341772294873097208, −9.065090904929816847781054423310, −8.409796115399264563671751933981, −7.42609238751831084620351594360, −5.53348027256752660465602480540, −4.38632761921007374570550890095, −3.63037610013973777395881448394, 1.66126434518535741943017728961, 2.76655547946686439723719666365, 4.48245514255955596853407628478, 6.43990826407911910927452873431, 7.69546574053836092130561842528, 8.656475991023175789593617662327, 9.414848825582525488123294553849, 11.40860102122929142696215116019, 11.68213889577964285256706086044, 12.82592186489966781436999236792

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