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Properties

Label 2-140-140.23-c1-0-19
Degree $2$
Conductor $140$
Sign $-0.524 + 0.851i$
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  + (1.18 − 0.764i)2-s + (−2.38 − 0.638i)3-s + (0.831 − 1.81i)4-s + (−0.525 − 2.17i)5-s + (−3.32 + 1.06i)6-s + (−2.10 + 1.60i)7-s + (−0.401 − 2.79i)8-s + (2.68 + 1.54i)9-s + (−2.28 − 2.18i)10-s + (4.09 − 2.36i)11-s + (−3.14 + 3.80i)12-s + (0.0592 + 0.0592i)13-s + (−1.27 + 3.51i)14-s + (−0.135 + 5.51i)15-s + (−2.61 − 3.02i)16-s + (4.77 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−1.37 − 0.368i)3-s + (0.415 − 0.909i)4-s + (−0.235 − 0.971i)5-s + (−1.35 + 0.433i)6-s + (−0.794 + 0.607i)7-s + (−0.142 − 0.989i)8-s + (0.893 + 0.515i)9-s + (−0.723 − 0.690i)10-s + (1.23 − 0.713i)11-s + (−0.907 + 1.09i)12-s + (0.0164 + 0.0164i)13-s + (−0.339 + 0.940i)14-s + (−0.0349 + 1.42i)15-s + (−0.654 − 0.755i)16-s + (1.15 + 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.524 + 0.851i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ -0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509780 - 0.912498i\)
\(L(\frac12)\) \(\approx\) \(0.509780 - 0.912498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.764i)T \)
5 \( 1 + (0.525 + 2.17i)T \)
7 \( 1 + (2.10 - 1.60i)T \)
good3 \( 1 + (2.38 + 0.638i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0592 - 0.0592i)T + 13iT^{2} \)
17 \( 1 + (-4.77 - 1.27i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.31 + 2.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.292 - 1.09i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.27iT - 29T^{2} \)
31 \( 1 + (-4.01 + 2.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.596 + 2.22i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 + (-1.57 + 1.57i)T - 43iT^{2} \)
47 \( 1 + (2.67 - 0.716i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.44 + 9.12i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.67 - 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.978 - 1.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.131 + 0.491i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (2.48 - 9.26i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.05 - 5.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.62 + 5.62i)T - 83iT^{2} \)
89 \( 1 + (-14.4 - 8.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.81 + 5.81i)T - 97iT^{2} \)
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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

−12.53276569132532500322316292280, −11.93325933809859549848561200757, −11.35425914435030242314746916764, −9.990358156256052254883390837029, −8.869558512200415988246399431640, −6.84745560435674768508069778518, −5.88753573432637506863298667958, −5.15436931990624916819673430089, −3.59564812655457191389960407807, −1.06837805019699597449568474699, 3.42741919658605946136204007863, 4.53995906911238604223144582898, 6.03369655538813553346183768983, 6.62819917933564965180361005140, 7.62597435546900247736553244837, 9.762013851194959437587344011536, 10.63091056783554984248051287521, 11.88256716248687467482129065845, 12.09241057052902108900440774613, 13.62435895007789363418506979105

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